What are imaginary numbers? At school, I really struggled to understand the concept of imaginary numbers.  My teacher told us that an imaginary number is a number that has something to do with the square root of $-1$.  When I tried to calculate the square root of $-1$ on my calculator, it gave me an error.  To this day I still do not understand imaginary numbers.  It makes no sense to me at all.  Is there someone here who totally gets it and can explain it?
Why is the concept even useful?
 A: One answer for why imaginary (and complex) numbers are useful is that they provide solutions to polynomial equations. (The square root of -1 part comes from trying to solve the equation $x^2 = -1$, which has no real number solutions.) The Fundamental Theorem of Algebra states that any polynomial equation with real (or even complex!) coefficients has solutions in the complex number system. 
The theorem doesn't always seem very powerful, because a lot of times we discard all non-real solutions. But, this isn't always the case. Linear (ordinary) differential equations can be solved by first solving an associated polynomial equation. The complex solutions to the polynomial equation end up influencing the solution to the differential equation. 
A: This is probably not helpful for someone first learning about imaginary numbers, but my personal motivation for complex numbers is so that every linear transformation over the reals can be decomposed into a direct sum of shift plus scaling operators, ie. the Jordan normal form exists.
If you work with matrices/linear operators over the reals for long enough, this is something that "feels" like it should be true - like some sort of linear algebra version of the pigeonhole principle - but it doesn't quite work over the reals because of rotation matrices. On the other hand, rotation is "like" a scaling because if you apply a rotation twice, it's the same as rotating twice as much once, so one feels this shouldn't really be an obstruction.
In any case, complex numbers are exactly the number system you need to ensure Jordan normal form exists, where rotations are scalings of complex eigenvectors by a complex number.
A: @  I just want to add that generally speaking, besides the whole numbers, none of the numbers we use "exist" in the real world.
@ Sachin Kainth
Hmmm. The question you ask is a deep one. The answer is far from easy. The quote above from one of the earlier answers is not right. I am not sure that "whole numbers" do "exist in the real world", let alone that "real numbers" do, or that complex numbers, quaternions or octonions don't.
The relationship between maths and the real world is extremely mysterious. It goes straight into the classic "God is a Mathematician" statement. I do not remotely have time to go into it properly here. Whole books have been written about it. Some better than others.
One viewpoint is simply to ignore questions of "reality" or relationship to the "real world" and say that complex numbers are exceedingly useful. Another approach is to go down the Clifford Algebra route, originally pioneered by William Clifford (1845-79) at my old college (Trinity, Cambridge) and which has recently seen an explosion of interest by theoretical physicists led perhaps by Stephen Gull at the Cavendish. Roger Penrose (Oxford) is also interesting on the subject of complex numbers.
But all that stuff requires some math sophistication to understand. An important prior question is "real numbers" or even fractions. There are many deeply puzzling and paradoxical questions about them. I suspect you have not been exposed to them.
Looking for someone who "totally gets it" is likely to be a vain hope. If you find them, let me know!
A: Let's go through some questions in order and see where it takes us. [Or skip to the bit about complex numbers below if you can't be bothered.]
What are natural numbers?
It took quite some evolution, but humans are blessed by their ability to notice that there is a similarity between the situations of having three apples in your hand and having three eggs in your hand. Or, indeed, three twigs or three babies or three spots. Or even three knocks at the door. And we generalise all of these situations by calling it 'three'; same goes for the other natural numbers. This is not the construction we usually take in maths, but it's how we learn what numbers are.

Natural numbers are what allow us to count a finite collection of things. We call this set of numbers $\mathbb{N}$.

What are integers?
Once we've learnt how to measure quantity, it doesn't take us long before we need to measure change, or relative quantity. If I'm holding three apples and you take away two, I now have 'two fewer' apples than I had before; but if you gave me two apples I'd have 'two more'. We want to measure these changes on the same scale (rather than the separate scales of 'more' and 'less'), and we do this by introducing negative natural numbers: the net increase in apples is $-2$.

We get the integers from the naturals by allowing ourselves to take numbers away: $\mathbb{Z}$ is the closure of $\mathbb{N}$ under the operation $-$.

What are rational numbers?
My friend and I are pretty hungry at this point but since you came along and stole two of my apples I only have one left. Out of mutual respect we decide we should each have the same quantity of apple, and so we cut it down the middle. We call the quantity of apple we each get 'a half', or $\frac{1}{2}$. The net change in apple after I give my friend his half is $-\frac{1}{2}$.

We get the rationals from the integers by allowing ourselves to divide integers by positive integers [or, equivalently, by nonzero integers]: $\mathbb{Q}$ is (sort of) the closure of $\mathbb{Z}$ under the operation $\div$.

What are real numbers?
I find some more apples and put them in a pie, which I cook in a circular dish. One of my friends decides to get smart, and asks for a slice of the pie whose curved edge has the same length as its straight edges (i.e. arc length of the circular segment is equal to its radius). I decide to honour his request, and using our newfangled rational numbers I try to work out how many such slices I could cut. But I can't quite get there: it's somewhere between $6$ and $7$; somewhere between $\frac{43}{7}$ and $\frac{44}{7}$; somewhere between $\frac{709}{113}$ and $\frac{710}{113}$; and so on, but no matter how accurate I try and make the fractions, I never quite get there. So I decide to call this number $2\pi$ (or $\tau$?) and move on with my life.

The reals turn the rationals into a continuum, filling the holes which can be approximated to arbitrary degrees of accuracy but never actually reached: $\mathbb{R}$ is the completion of $\mathbb{Q}$.

What are complex numbers? [Finally!]
Our real numbers prove to be quite useful. If I want to make a pie which is twice as big as my last one but still circular then I'll use a dish whose radius is $\sqrt{2}$ times bigger. If I decide this isn't enough and I want to make it thrice as big again then I'll use a dish whose radius is $\sqrt{3}$ times as big as the last. But it turns out that to get this dish I could have made the original one thrice as big and then that one twice as big; the order in which I increase the size of the dish has no effect on what I end up with. And I could have done it in one go, making it six times as big by using a dish whose radius is $\sqrt{6}$ times as big. This leads to my discovery of the fact that multiplication corresponds to scaling $-$ they obey the same rules. (Multiplication by negative numbers responds to scaling and then flipping.)
But I can also spin a pie around. Rotating it by one angle and then another has the same effect as rotating it by the second angle and then the first $-$ the order in which I carry out the rotations has no effect on what I end up with, just like with scaling. Does this mean we can model rotation with some kind of multiplication, where multiplication of these new numbers corresponds to addition of the angles? If I could, then I'd be able to rotate a point on the pie by performing a sequence of multiplications. I notice that if I rotate my pie by $90^{\circ}$ four times then it ends up how it was, so I'll declare this $90^{\circ}$ rotation to be multiplication by '$i$' and see what happens. We've seen that $i^4=1$, and with our funky real numbers we know that $i^4=(i^2)^2$ and so $i^2 = \pm 1$. But $i^2 \ne 1$ since rotating twice doesn't leave the pie how it was $-$ it's facing the wrong way; so in fact $i^2=-1$. This then also obeys the rules for multiplication by negative real numbers.
Upon further experimentation with spinning pies around we discover that defining $i$ in this way leads to numbers (formed by adding and multiplying real numbers with this new '$i$' beast) which, under multiplication, do indeed correspond to combined scalings and rotations in a 'number plane', which contains our previously held 'number line'. What's more, they can be multiplied, divided and rooted as we please. It then has the fun consequence that any polynomial with coefficients of this kind has as many roots as its degree; what fun!

The complex numbers allow us to consider scalings and rotations as two instances of the same thing; and by ensuring that negative reals have square roots, we get something where every (non-constant) polynomial equation can be solved: $\mathbb{C}$ is the algebraic closure of $\mathbb{R}$.

[Final edit ever: It occurs to me that I never mentioned anything to do with anything 'imaginary', since I presumed that Sachin really wanted to know about the complex numbers as a whole. But for the sake of completeness: the imaginary numbers are precisely the real multiples of $i$ $-$ you scale the pie and rotate it by $90^{\circ}$ in either direction. They are the rotations/scalings which, when performed twice, leave the pie facing backwards; that is, they are the numbers which square to give negative real numbers.]
What next?
I've been asked in the comments to mention quaternions and octonions. These go (even further) beyond what the question is asking, so I won't dwell on them, but the idea is: my friends and I are actually aliens from a multi-dimensional world and simply aren't satisfied with a measly $2$-dimensional number system. By extending the principles from our so-called complex numbers we get systems which include copies of $\mathbb{C}$ and act in many ways like numbers, but now (unless we restrict ourselves to one of the copies of $\mathbb{C}$) the order in which we carry out our weird multi-dimensional symmetries does matter. But, with them, we can do lots of science.
I have also completely omitted any mention of ordinal numbers, because they fork off in a different direction straight after the naturals. We get some very exciting stuff out of these, but we don't find $\mathbb{C}$ because it doesn't have any natural order relation on it.
Historical note
The above succession of stages is not a historical account of how numbers of different types are discovered. I don't claim to know an awful lot about the history of mathematics, but I know enough to know that the concept of a number evolved in different ways in different cultures, likely due to practical implications. In particular, it is very unlikely that complex numbers were devised geometrically as rotations-and-scalings $-$ the needs of the time were algebraic and people were throwing away (perfectly valid) equations because they didn't think $\sqrt{-1}$ could exist. Their geometric properties were discovered soon after.
However, this is roughly the sequence in which these number sets are (usually) constructed in ZF set theory and we have a nice sequence of inclusions
$$1 \hookrightarrow \mathbb{N} \hookrightarrow \mathbb{Z} \hookrightarrow \mathbb{Q} \hookrightarrow \mathbb{R} \hookrightarrow \mathbb{C}$$
Stuff to read


*

*The other answers to this question give very insightful ways of getting $\mathbb{C}$ from $\mathbb{R}$ in different ways, and discussing how and why complex numbers are useful $-$ there's only so much use to spinning pies around.

*A Visual, Intuitive Guide to Imaginary Numbers $-$ thanks go to Joe, in the comments, for pointing this out to me.

*Some older questions, e.g. here and here, have some brilliant answers.


I'd be glad to know of more such resources; feel free to post any in the comments.
A: Imaginary numbers can also be thought of as a simple hack mathematicians use when they want to keep units separate.
Need a result with more than one component? make it a multiple of something that won't resolve. Pretty handy.
A: You ask why imaginary numbers are useful. As with most extensions of number systems, historically such generalizations were invented because they help to simplify certain phenomena in existing number systems. For example, negative numbers and fractions permit one to state in a single general form the quadratic equation and its solution (older solutions bifurcated into many cases, avoiding negative numbers and fractions). One of the primary reasons motivating the invention of complex numbers is that they serve to linearize what would otherwise be nonlinear phenomena - thus greatly simplifying many problems. Here are some examples. 
Consider the problem of representing integers as sums of squares $\rm\: n = x^2 + y^2$. Early solutions to this and related problems employed a complicated arithmetic of binary quadratic forms. Such arithmetic was quite intricate and often very nonintuitive, e.g. even the proof of associativity of composition of such forms was a tour de brute force, occupying pages of unmotivated computations in Gauss' Disq. Arith. But this quadratic arithmetic of binary quadratic forms can be linearized. Indeed, by factorization $\rm\: x^2 + y^2 = (x+y{\it i})(x-y{\it i}),$ which allows us to view sums of squares as norms of Gaussian integers $\rm\:x+y{\it i},\ \ x,y\in \Bbb Z.\:$ But just like the rational integers $\Bbb Z,$ these "imaginary" integers have a Euclidean algorithm, so enjoy unique factorization into primes. By considering all the possible factorizations of $\rm\:n\:$ in the Gaussian integers we obtain all the possible representations of $\rm\:n\:$ as a sum of squares. In a similar way, "rational, real" arithmetic of integral quadratic forms becomes much simpler by passing to the "irrational" and/or "imaginary" arithmetic of quadratic number fields. This line of research led to the discovery of ideals and modules, fundamental linear structures at the heart of modern number theory and algebra. [See this answer and its links for a more precise description of the equivalence between quadratic forma and ideals].
Thus, by factorizing completely over $\Bbb C$, we have reduced the complicated nonlinear arithmetic of binary quadratic forms to the simpler, linear arithmetic of Gaussian integers, i.e. to the more familiar arithmetical structure of a unique factorization domain (in fact a Euclidean domain).  Analogous linearization serves to simplify many problems. For example, when integrating or summing  rational functions (quotients of polynomials), by factoring denominators over $\Bbb C$ (vs. $\Bbb R)$ and taking partial fraction decompositions, the denominators are at worst powers of linear (vs. quadratic) polynomials - which greatly simplifies matters. More generally, when solving constant coefficient differential or difference equations (recurrences), by factoring their characteristic (operator) polynomials over $\Bbb C,$ we reduce to solutions of linear (vs. quadratic) differential or difference equations. In the same way, there are many real problems (over $\Bbb R)$ whose simplest solutions are obtained by an imaginary detour (over $\Bbb C).$  Perhaps readers will mention more such problems in the comments.
A: I just think of imaginary numbers as a definition. In the "real world" you cannot take the square root of $−1$ (which is what is happening with your calculator). However, we just define some "number", call it $i$, such that $i^2=−1$, add it to our number system and see what happens. So when you study imaginary numbers, you are just "seeing what happens".
One can then write every number as $a+ib$ where $a,b\in\mathbb{R}$ ($a$ and $b$ are real numbers) and $i^2=−1$. In his comment, ivan is taking this pair $(a,b)$ and pointing out that this pair defines a point on a plane (so, like, a piece of paper, as when you draw a graph). This is the way that people often view imaginary numbers - as points on the plane (and the plane is the Complex Plane, or an Argand diagram).
A: Imaginary numbers were invented to make calculations easier. Everyone knows the quadratic formula; when Cardano was working on the formula for cubics (known as Cardano's formula), he found out that it was extremely hard to write down a formula unless you out down some symbol as a placeholder for $\sqrt{-1}$, which you manipulate like a number and which always cancelled out in the end. So he left it in. He was embarrassed by it, and called it imaginary, but the formula worked. Mathematicians later found out that imaginary numbers made a lot of formulas easier, like finding a formula for $\sin(3x)$, and so they found consistent rules for them. Ever since then, they've kept making formulas easier.
A: You can get imaginary and complex numbers with your calculator. Switch to complex mode (press the mode button and sift through the modes until you see something like CMPLX) and select it. Now input the square root of -1 again. You should get the answer i. An imaginary number is simply i with a coefficient in front, e.g. i (1i), 2i, 3i, 1/6i, 345i, sqrt3i, or whatever.
The concept is useful because you need it to pass Maths... Sometimes, for example, you may be asked to give the complex roots of a quadratic equation if the discriminant is negative.
A: It's an extension field . . . but since you probably don't know that, the terminology is horrible!  Just think of imaginary numbers as the completion of real numbers so that you can find solutions to the equation
$x^2 + 1 = 0.$
If you set $i = \sqrt{-1}$, then $i$ and $-i$ are solutions to this polynomial.  There are no 'real solutions'.  
It is known that any univariate system of degree $n$ has exactly $n$ solutions in the complex plane.  This can be generalized further to multivariate square systems in that the max number of solutions is the multiplication of the largest degree of each function.
For example,
$x_1^3 + x_1*x_2 + 4 = 0$
$x_1^2*x_2 + x_1 + x_2 - 2 = 0$
has (at most) $3*4 = 12$.  This is known as Bezout's Theorem and is a result of classical algebraic geometry.  Letting $i = \sqrt{-1}$ is necessary to find all the solutions (and it is possible to find these solutions numerically, up to an arbitrary choice of precision).
A: Let's add something which hasn't quite been said yet. Suppose I take polynomials - I could be using rational coefficients, or real coefficients, for example. Because polynomials have a division algorithm, for any given polynomial $p(x)$, I can write $$p(x)=(x^2+1)q(x)+p^*(x)$$ where $p^*(x)$ is a linear polynomial, of the form $ax+b$. I want to know what happens when I replace $p(x)$ with $p^*(x)$, and whether I can do this consistently.
As an example $2x^3+4x^2+3x+1=(x^2+1)(2x+4)+x-3$
I can confirm, for example, that $$p_1(x)p_2(x)=(x^2+1)\left((x^2+1)q_1(x)q_2(x)+p^*_1(x)q_2(x)+p^*_2(x)q_1(x)\right)+p^*_1(x)p^*_2(x)$$So reducing to the linear remainder is compatible with multiplication (though the product of remainders may have to be reduced). And it is easy too to see that this works for addition. If we are not worried about multiples of $x^2+1$ we can produce a simpler polynomial arithmetic, where the results are always linear.
What use is this? Well note that $$x^2=(x^2+1)-1$$ so the linear equivalent of $x^2$ is simply $-1$ - and we have produced a simplified form of polynomial arithmetic in which it makes sense to say $x^2=-1$. Of course we haven't said what $x$ is at any stage, so we haven't yet said it is a number of any kind. We can, however, call any multiple of $x$ in this system an "imaginary number" if we so wish.
The geometric explanations are definitely the ones to go for first. But the more abstract algebraic approach becomes very significant in more advanced work. And the abstract approach shows that we can calculate in this way if it happens to be useful - we have a consistent system. And it has proven to be very useful indeed, not least because (this is a geometric insight) it enables us to treat the points on the unit circle as numbers - an insight which transforms the way in which we think about periodic functions and waves.
A: I went to school for electrical engineering ($7$ years total) and we used imaginary numbers all over the place.
Even with all that schooling, this is probably the clearest explanation of imaginary numbers I've seen:
A Visual, Intuitive Guide to Imaginary Numbers.
HTH.
A: As no one has mentioned about this, here is an excellent intuitive video series (that shouldn't be missed) by Welch Labs as Imaginary Numbers are Real. 
These are a set of 13 videos that beautifully help with the visualization of complex numbers from A Simple Complex Plane, all the way up to Riemann Surfaces. Here is something from the videos to get you interested.
Consider a Function $f(x)=x^{2}+1$.
Here is the plot of the function in the real x-y plane: 

(Source: WolframAlpha)
Now according to the Fundamental Theorem of Algebra we should have n-roots for n-th degree polynomial but if you consider the graph for the given function it doesn't appear to intersect the x-axis right ?
Well, the thing is, we are not seeing it correctly and have not included a fundamental set of numbers : Complex Numbers which have both real and imaginary part but don't get confused yet as both the parts are quite real.The below GIF from the above videos beautifully plots the the function in the Complex plane (The vertical axis that comes out of the paper being the imaginary axis).

(Source: Welch Labs)
So you can see that the function actually intersects the x-axis but in a different dimension and that is the imaginary plane. This gives an excellent intuitive sense of why imaginary numbers are very real! I would recommend the users to view all the videos as there's a lot of graphic animations for better visualization and understanding.
A: Well, as you know there's no real number whose square is negative. But now imagine numbers which are. Let's call them imaginary. Now what properties would such numbers have? Well, there would be for example a number whose square is $-1$. Let's call that number the imaginary unit and give it the name $\mathrm i$. Now if we multiply this number with some real number, that is, use $r\mathrm i$, we get a number whose square is $(\mathrm ir)^2 = \mathrm i^2r^2 = -r^2$. Since all positive numbers can be written as $r^2$, we get that all negative numbers can be written as $(\mathrm ir)^2$. Thus the products $\mathrm ir$ are our imaginary numbers. We also see that $(-\mathrm i)^2 = (-1)^2\mathrm i^2 = -1$, so there are actually two numbers whose square is $-1$ (which makes sense because, after all, there are also two numbers whose square is $1$, namely $1$ and $-1$).
OK, but what happens if we add a real number and one of out imaginary numbers. Well, now things get complex. We get general complex numbers.
OK, but how do we know that we've not just made some nonsense, similar to the nonsense that we get when we invent a number $o$ so that $0o=1$? Well to see that, we recognize that all complex numbers are of the form $x+\mathrm iy$ with real numbers $x$ and $y$, and thus the pair $(x,y)$ completely specifies a complex number. Therefore now we re-derive the complex numbers as pairs of real numbers, but now using proper mathematical instruments so we know for sure that whatever we do is well defined. Since doing that we arrive at the very same structure which we just had derived in a quite informal way, we know that the complex numbers are a sound mathematical structure.
OK, now that we have invented the imaginary and complex numbers, are they useful for something? Well, indeed they are. For example, several mathematical statements are much easier in complex numbers than in real numbers. For example, with complex numbers, every polynomial can be written in the form $a(x-x_1)(x-x_2)\cdots(x-x_n)$. With real numbers, this is impossible for polynomials having for example factors of the form $(x^2+1)$. Moreover, we have the very useful relation $\mathrm e^{\mathrm i\phi} = \cos\phi + \mathrm i\sin\phi$. So forget about complicated addition theorems for sine and cosine. Just rewrite your formula in complex exponentials and enjoy the simple relation $\mathrm e^{\mathrm i(\alpha+\beta)}=\mathrm e^{\mathrm i\alpha}\mathrm e^{\mathrm i\beta}$.
Finally, if you want to do quantum physics (and almost all modern physics is quantum physics) you'll find that you have to use complex numbers.
A: The term "imaginary" is somewhat disingenuous. It's a real concept, with real (at least theoretical) application, just like all the "real" numbers.
Think back to that algebra class. You were asked to solve a polynomial equation; that is, find all the values of X for which the entire equation evaluates to zero. You learned to do this by polynomial factoring, simplifying the equation into a series of first-power terms, and then it was easy to see that if any one of those terms evaluated to zero, then everything else, no matter its value, was multiplied by zero, producing zero.
You tried this on a few quadratic equations. Sometimes you got one answer (because the equation was $y=ax^2$ and so the only possible answer was zero), sometimes you got two (when the equation boiled down to $y= (x\pm n)(x \pm m)$, and so when $x=-m$ or $x=-n$ the equation was zero), and a couple of times, you got no answers at all (usually, an equation that breaks down to $y=(x+n)(x+m)$ doesn't evaluate to zero at $x=-m$ or $x=-n$).
In your algebra class, you're told this just happens sometimes, and the only way to make sure any factored term $(x\pm k)$ represents a real root is to plug in $-k$ for $x$ and solve. But, this is math. Mathematicians like things to be perfect, and don't like these "rules of thumb", where a method works sometimes but it's really just a "hint" of where to look. So, mathematicians looked for another solution.
This leads us to application of the quadratic formula: for $ax^2 + bx + c = 0$, $x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$. This formula is quite literally the solution of the general form of the equation for x, and can be derived algebraically. We can now plug in the coefficients, and find the values of $x$ where $ax^2 + bx + c=0$. Notice the square root; we're first taught, simply, that if $b^2-4ac$ is ever negative, then the roots you'd get by factoring the equation won't work, and thus the equation has no real roots. $b^2-4ac$ is called the determinant for this reason. 
But, the fact that $b^2-4ac$ can be negative remains a thorn in our side; we want to solve this equation. It's sitting right in front of us. If the determinant were positive, we would have solved it already. It's that pesky negative that's the problem.
Well, what if there was something we could do, that conforms to the rules of basic algebra, to get rid of the negative? Well, $-m = m*-1$, so what if we took our term that, for the sake of argument, evaluated to $-36$, and made it $36*-1$? Now, because $\sqrt{mn} = \sqrt{m}\sqrt{n}$, $\sqrt{-36} = \sqrt{36}\sqrt{-1} = 6\sqrt{-1}$. We've simplified the expression by removing what we can't express as a real number from what we can. 
Now to clean up that last little bit. $\sqrt{-1}$ is a common term whenever the determinant is negative, so let's abstract it behind a constant, like we do $\pi$ and $e$, to make things a little cleaner. $\sqrt{-1} = i$. Now, we can define some properties of $i$, particularly a curious thing that happens as you raise its power:
$$i^2 = \sqrt{-1}^2 = -1$$
$$i^3 = i^2*i = -i$$
$$i^4 = i^2*i^2 = -1*-1 = 1$$
$$i^5 = i^4*i = i$$
We see that $i^n$ transitions through four values infinitely as its power $n$ increases, and also that this transition crosses into and then out of the real numbers. Seems almost... circadian, rotational. As Clive N's answer so elegantly explains it, that's what imaginary numbers represent; a "rotation" of the graph through another plane, where the graph DOES cross the $x$-axis. Now, it's not actually really a circular rotation onto a new linear z-plane. Complex numbers have a real part, as you'd see by solving the quadratic equation for a polynomial with imaginary roots. We typically visualize these values in their own 2-dimensional plane, the complex plane. A quadratic equation with imaginary roots can thus be thought of as a graph in four dimensions; three real, one imaginary.
Now, we call $i$ and any product of a real number and $i$ "imaginary", because what $i$ represents doesn't have an analog in our "everyday world". You can't hold $i$ objects in your hand. You can't measure anything and get $i$ inches or centimeters or Smoots as your result. You can't plug any number of natural numbers together, stick a decimal point in somewhere and end up with $i$. $i$ simply is.

As far as having use outside "ivory tower" math disciplines, a big one is in economics; many economies of scale can be described as a function of functions of the number of units produced, with a cost term and a revenue term (the difference being profit or loss), each of these in turn defined by a function of the per-unit sale price or cost and the number produced. This all generally simplifies to a quadratic equation, solvable by the quadratic formula. If the roots are imaginary, so are the breakeven points (and your expected profits).
Another good one is in visualizations of complex numbers, and of their interactions when multiplied. The first one I was exposed to is a well-known series set, produced by taking an arbitrary complex number, squaring it ($(a+bi)^2 = (a+bi)(a+bi) = a^2 + 2abi + b^2i^2 = a^2-b^2 + 2abi$), and then adding back its original value. Repeated to infinity with this number, the series either converges to zero or diverges to infinity (with a few starting numbers exhibiting periodicity; they'll jump around infinitely between a finite number of points much like $i$ itself does). The set of all complex numbers for which the series does not diverge is the Mandelbrot set or M-set, and while the area of the graph is finite, its perimeter is infinite, making the graph of this set a fractal (one of the most highly-studied, in fact). 
The Mandelbrot set can in turn be defined as the set of all complex numbers $c$ for which the Julia set $J(f)$ of $f(z)=z^2 + c \to z$ is connected. A Julia set exists for every complex polynomial function, but usually the most interesting and useful sets are the ones for values of $c$ that belong to the M-set; Julia fractals are produced much the same way as the M-set (by repeated iteration of the function to determine if a starting $z$ converges or diverges), but $c$ is constant for all points of the set instead of being the original point being tested. You can define Julia sets with all sorts of fractal shapes. These fractals, more accurately the iterative evaluation behind them, are used for pseudorandom number generation, computer graphics (the sets can be plotted in 3-d to create landscapes, or they can be used in shaders to define complex reflective properties of things like insect shells/wings), etc.
A: This question has already been answered quite thoroughly, but I just want to add that generally speaking, besides the whole numbers, none of the numbers we use "exist" in the real world.  The only reason we have adopted extensions to the whole numbers to the natural, integer, rational, real, and complex sets in turn is because these extensions make problems solvable when thinking abstractly.  At the end of the day, everything relates back to the whole numbers, however.
Most people use all of the sets except for complex numbers in very commonplace, everyday situations, which is why we've come to view everything up to the real numbers as being fairly intuitive, at least at first glance. (When you dig under the surface, everything gets a great deal more subtle, which is why there are people who study primarily numbers, who we call number theorists.  But that's a whole other story.)
It's important to note that this progression isn't the only way to extend the whole numbers.  There are hundreds of different arithmetics that have been designed, many not even based on the whole numbers.  It's just that the usual extension applies to so many situations that come up commonly.  (People who study universal algebra study the ways in which different possible math systems are alike and different.  But that's a whole other story as well.)
Complex numbers have taken their place as the normal extension to the reals because they are so useful when dealing with polynomials, which happen to arise in a massive number of mathematical situations.  They also allow the exponential function and the trigonometric functions to be viewed as special cases of the same thing, through Euler's Formula, which enables all sorts of great algebra tricks.  Specifically, these sorts of functions pop up constantly when using either Taylor or Fourier series to simplify the process of working on problems with tricky transcendental functions.  Complex numbers make dealing with these representations a breeze (relatively).
There are even further extensions.  If instead of worrying about how to take the square root of -1, you worry about what happens past infinity, the real numbers can alternatively be expanded in several jumps to include hyperreals, superreals, and surreals.  None of these systems have caught on, though, because we have alternative ways of dealing with the infinite and infinitesimal quantities in calculus that people find more powerful/convenient.
You can also zip on past complex numbers to Quaternions, and octonions on top of that.  Vectors generalize all of the above.  They aren't often though of as numbers, but are similar in that they generalize the concept of a property of an object having a mathematical value.  Matrixes generalize vectors, and tensors generalize matrixes.  
As you climb this ladder, you gain more and more mathematical power, but you start to lose properties that we expect of whole numbers.  For complex numbers, order (greater than/less than) begins to become ambiguous.  We generally don't think of vectors as "numbers" because we want all operations on vectors to work regardless of dimension, and most of the arithmetic operations don't really generalize.  With matrixes, the commutative property goes out the window, and things start to get really weird, especially when the matrixes aren't square.  And so forth.
All of this to say that numbers are best viewed as machinery.  Different number systems are really only used to the extent which they make a given math situation or problem easier to think about.  If you're an engineer, complex numbers do this in many, many situations, which justifies their added...complexity.  If you're not an engineer, they're definitely worth understanding, but you may not find uses for them on a daily basis.  
A: Complex numbers are just a handy way to handle two dimensional points and move them around. The key to it is understanding that i × i = −1 is just a simple by-product of moving these points around.
Real numbers correspond to numbers on a line (one dimension), which is usually how they are represented: a single axis where each number has a position.
Operations on these real numbers have been defined to apply the two most basic transformations:


*

*Translation (addition)
Move a point by a given amount.

*Scaling (multiplication)
Move a point by an amount related to its value, e.g., two times further than it was.
Now, for a number of situations, you need to handle elements that are not on a line, but on a plane—you are now in two dimensions. When working in two dimensions, you need to know where you are horizontally and vertically, which you usually represent with two numbers. For instance, (3, 2) is “3 to the right, 2 up”. Complex numbers are designed to manipulate these elements with dimensions with “simple” mathematics.
We define i as being the vertical unit. 2i is “2 up”, −4i is “4 down”, and 3 + 2i is “3 to the right and 2 up”. We still can use translation and scaling like in the one-dimension case, but we would like to add something: rotation. How do I turn “2 to the right” into “2 up”?
The solution comes with multiplying by i. If 1 is “1 to the right” and 1 × i is “1 up”, then it means that multiplying by i is simply rotating by 90 degrees with point 0 as a center, counter-clockwise. 2 × i = 2i means “2 to the right” multiplied by i gives “2 up”.
And this is where it gets interesting: rotating the point “1 to the right” by 90 degrees gives “1 up”. Rotating it again by 90 degrees gives “1 left”. This means that multiplying 1 twice by i gives −1.
We have 1 × i × i = −1, and since i × i = −1, i is by definition the square root of −1.
A: This argument is a loose argument for the sake of simplicity and because I know little about the subject. However, I think it may be good for non-mathematicians. 
The simplistic view is to note that imaginary numbers (or Complex Numbers) are numbers that are defined by humans to describe quantities different from the numbers we use in our day-to-day life (unless you are a scientist). They have certain rules that are somewhat different than those we use to calculate with non-complex numbers. Hence, the subjects of (Complex Variables and Complex Analysis). 
In mathematics, this is not strange. There are concepts that may look surprising until you study them carefully. For example, in Binary Numbers $1+1=10$. This result does not make any sense unless you understand and realize that the result is valid in the Binary System, domain or framework. 
Personally, I thought about this before I read your question, and found that the problem comprehending such concepts could arise when you think about a concept outside its framework (or domain) and try to rationalize the results using our every day concepts. 
For example trying to evaluate the $\sqrt{-1}$ on a regular calculator with no setting for Imaginary Arithmetic (the proper name is probably Complex Arithmetic). The calculator has to be set to the correct mode (or framework) to give a correct result. In fact, the software in your calculator should have given you a decent error message (or better yet the result of $i$ with a warning note).
Again, the same thing will happen if you are using your calculator in Binary mode to add $1+1$, you will not get the familiar $2$.
Many other examples can be driven around the same concept.
I hope this helps.
A: I think of i as just a symbol to represent an operation 

      √-1

When we want the square root of -1, just represent the whole statement with a symbol without evaluating it. This avoids the necessity of trying to explain it further, we don't need to map the answer to some real world concept, it's just a saved operation. We also know that the square root has the following property: 

      √x √x = x

No matter what the x. i.e. 

      √-1 √-1 = -1

Numbers are useful to me when they represent concepts in the real world. I don't map i to anything in the real world but with this ability to represent the operation, I can now manipulate it in algebraic expressions to ultimately get back to non-imaginary numbers that I do find useful. http://en.wikipedia.org/wiki/Euler%27s_formula
A: Check it out, I just learned this very recently:
Define the set of all ordered pairs $(x, y)$, call it $\mathbb{C}$, the set of complex numbers. We call $x$ the real part, and y the imaginary part.
Now define multiplication like this:
$(x, y) \cdot (a,b) = (xa-yb, xb+ya) $
Now I'm not sure what that's supposed to be but observe:
$(0, 1)^2 = (0,1) \cdot(0,1)= (0-1, 0+0) = (-1, 0)$
Since the second number in the ordered pair is the imaginary part, (0,1) corresponds to $0+1\cdot i =i$. (In fact all complex numbers $(x, y)$ correspond to $x+yi$ ).
So I have just shown you how defining multiplication that way results in $i^2 = -1$.
But that multiplication isn't the multiplication I'm familiar with!, you say.
Well guess what:
$a\cdot b = (a, 0) \cdot (b, 0) = (ab-0,0+0) = (ab, 0) = ab $
Yes it is!
So what I get from this is that essentially someone said: "What if there was a number that could be squared to get -1", and there you have it!
In fact, once you define addition like this:
$(a, b) + (x, y) = (a+b, x+y)$
I'm pretty sure you'll find this new system of complex numbers, $\mathbb{C}$, to be compatible with the old set of real numbers, $\mathbb{R}$ .
A: I'm surprised that, as far as I can see, no one has mentioned Paul Nahin's book "An imaginary tale : the story of √-1", pub: Princeton University Press ISBN 0-691-12798-0. It is a historical account of how  √-1 became a necessary mathematical tool, and is written in an easy to read conversational style. I keep re-reading parts of it, like going over old ground again with a friend.
Two reviews give contrasting opinions: the first very favourable http://plus.maths.org/content/imaginary-tale; the second giving a long list of (alleged -- I haven't checked them independently) inaccuracies and omissions: http://www.ams.org/notices/199910/rev-blank.pdf
A: 'What are imaginary numbers?'
The most honest answer I can think of is: we don't know.
But they are a very, very useful tool, as with many other mathematical abstractions that we really don't fully philosophically understand, and yet they are the true foundations of our modern civilizations, because they are practical and they proven to work very well.
A simple reasoning (I often give to my kid) goes as follows: what is the number that multiplied by itself gives 4? The answer is 2, or minus 2, of course. Then, what is the number that multiplied by itself gives 1? The answer is 1, or minus 1.
Now, what is the number that multiplied by itself gives 2? The answer for this one is not so simple (if you are not allowed to use ellipsis), its $\sqrt{2}$, or $-\sqrt{2}$. We have to start using more complicated symbols for this one now. Square root of two is not the most irrational number we can think of, as with $\pi$ for example, but it is so difficult to declare it on a finite character sequence basis that we have to resort to a simpler, and yet more obscure, symbol or set of symbols.
But, now, what is the number that multiplied by itself gives $-1$? We don't know! And perhaps we will never know. So, we decide to encapsulate all of our ignorance by naming it $i$, and it all works very well by only dealing with this symbol. It is an unknown, but it serves us very duly.
