Why would I want to multiply two polynomials? I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together? 
I flipped through some algebra books and have googled around a bit, and whenever they introduce polynomial multiplication they just say 'Suppose you have two polynomials you wish to multiply', or sometimes it's just as simple as 'find the product'. I even looked for some example story problems, hoping that might let me in on the secret, but no dice. 
I understand that a polynomial is basically a set of numbers (or, if you'd rather, a mapping of one set of numbers to another), or, in another way of thinking about it, two polynomials are functions, and the product of the two functions is a new function that lets you apply the function once, provided you were planning on applying the original functions to the number and then multiplying the result together. 
Elementary multiplication can be described as 'add $X$ to itself $Y$ times', where $Y$ is a nice integer number of times. When $Y$ is not a whole number, it doesn't seem to make as much sense.
Any ideas?
 A: It's possible to write down the set of matches between two strings as the result of multiplying two polynomials constructed from the strings. This turns out to be very useful: indeed, fast algorithms for string matching exploit the fast Fourier transform method used to multiply polynomials. 
To see how this works, think of a binary string as representing coefficients of a polynomial. Then matches between the strings correspond to nonzero coefficients of the product of the polynomial from one string and the reverse of the polynomial from the second. Here, the "multiplication" is over the AND-OR field, rather than the usual integer or GF(2) field. 
A: Multiplication of polynomials has a very nice combinatorial interpetation. 
Take a simple example: $1+x=x^0+x^1$. If you perform an experiment once, that experiment can either succeed or fail. You can fail the experiment in only one way in one performance, or you can succeed in one way. This is precisely encoded in the coefficients of the polynomial.
Look now at the product: $(1+x)\times(1+x)=1\times x^0+2\times x^1+1\times x^2$. The coefficients now encode in how many ways you can fail twice, how many ways you can succeed once and fail once and how many ways you can succeed twice.
So, I guess you get the trick now, $(1+x)^n$ will have encoded in its $k$th coefficient in how many ways you can succeed $k$ times but fail $n-k$ times.
This can be generalized to all kinds of polynomials.
A: If you were working with a polynomial ring, you'd definitely want to be able to multiply polynomials together. This can allow you to use power series to solve recurrences with generating functions.
Polynomial rings are also used in some cryptographic systems.
In a few years, you'll take calculus, and then you'll need to multiply polynomials in order to take derivatives of rational functions, to give one example.
The algebra you learn now will be very useful to your mathematics education later on. You want to get this stuff down cold now, so that later on you can focus on more complicated subjects.
A: When you take calculus, you will need to factor a polynomial p as a product of two polynomials a and b.  If you know how polynomial multiplication works, then finding factorizations is easier.  Learn how to multiply now so that you can factor easily later. :)
A: I think a better metaphor for multiplication comes from physics.  If your speed s is constant over some time t, then your distance traveled is just s times t.  Whatever fraction of time you travel, you will expect to have traveled some distance.  So even fractional multiplication makes some sense.
In the regular world, it's very common to use a polynomials (often linear ones) to approximate things that are changing.  For instance, if you are a grocer and you are trying to predict how many tomatoes you will sell tomorrow, it's not a bad approach to guess it will be the same as yesterday.  It's also not a bad approach to guess it might be the average of the last two days.  Or, you might come up with a more complicated formula involving the last 50 days.  All of these attempts to guess will have an error.  This is the point of studying these things, to do a better job and make smaller errors.  (For reasons, I can't go into, the squares and cubes of values can sometimes help make approximations more accurate.)
My point is that any quantity is a potential polynomial.  You might approximate your customers per day as a polynomial of the number of customers of the last two days.  You might approximate the value of sales per customer as another polynomial.  Well, if you wanted to know your potential income, customers times sales, you might multiply them.
I use polynomials all the time.  For instance, when I was helping a friend lose weight, I approximated their weight loss behavior using a polynomial.  It was fairly accurate.
So, keep with polynomials, they were worth it in the long run!
A: Suppose you want to make error-correcting codes (to do things like communicating with spacecraft, manufacturing CD's, and designing computer memory systems). Encoding and decoding BCH codes (as well as many other error correcting codes) involves multiplying polynomials.
Of course, nowadays the best error correcting codes are polar codes, turbo codes, or LPDC codes, which aren't based on polynomials. But for nearly 40 years many digital communications systems were based on multiplying polynomials.
A: In terms of simply the math, it is essentially a skill you need to learn. Mathematics becomes (a long with many other sciences) more abstract as you go up in rank, so to speak. You can think of multiplying and adding polynomials as the next logical step after learning how to multiply decimals.
There is a natural progression, that is both historic and customary to progress from the most concrete to the more abstract. In grade school, the first mathematical concept you learned is numbers, shortly thereafter you were introduced to how you could combine two numbers to form a third one (namely through addition and multiplication). After being introduced to natural numbers, the concept of combining two numbers was expanded to something called fractions, which also introduced the concept of division (the opposite of multiplication, if you will). After getting comfortable with fractions, you were introduced to decimals and irrational numbers, and the rules for combining and manipulating those numbers. At this point, mathematical education turns to more abstract concepts, the idea of functions and variables. You spend plenty of time (depending on how far you go in math, maybe the rest of your mathematical career) studying functions, how they behave, how to manipulate them. It turns out that polynomials, are some of the simplest functions you can think of, so it is only natural that you start with them. Following the trend as before, after understanding what functions (specifically polynomials) are, a natural next step is considering how you can combine those polynomials to make new polynomials. Later on in math, you will be introduced with even more ways of combining and manipulating functions to create new functions. This is where you are right now, you are learning to manipulate and combined some (if you think about it) incredibly abstract concept, you are learning to manipulate an object that took thousands of years of human thought to finalize. Of course, in reality this is just a small stepping stone to the greener pastures ahead, assuming you go on in mathematics you will encounter some of the most mind boggling abstract concepts (some of the negatively voted answers to this question will give you just a small hint of whats out there). You will find that the very simple concepts of addition and multiplication (and their respective inverses) will go a long way in the study of mathematics, but they are stripped of all their cuddly original meaning, to become the most beefed up behemoths of abstract mathematical thought.
Now, enough with the theoretical "its the way it should be". Lets move on to practicalities. Applications of numbers, are easy enough to see, addition,multiplication, subtraction, and division of numbers is an everyday thing. Considering how to apply polynomials to your everyday life (especially why you would ever think of combining polynomials) requires a little greater stretch of the mind (or a little more mathematics :D ). Without appealing to calculus (which is certainly an easy cop-out).
Polynomials, Describe a large variety of every day processes and events. The most common being the quadratic polynomial (this is due to a very deep physical fact, but I promised to avoid calculus). The free fall trajectory of an object in a gravitational field is described (neglecting friction) by a quadratic polynomial. Even in simple physical problems you can see the multiplication of monomials and quadratics. For example, solving for when a shot cannon hits the ground is merely the root of the polynomial, conversely, if you know where the canon hit the ground you can multiply to monomials $(x-r)$ where $r$ is the root, to get the trajectory. 
If we deviate a little from the math you have been exposed to, there are more abstract polynomials called Laguerre Polynomials that are present in the solution to the hydrogen atom. So if you (for some reason) wanted to calculate the shape of a 3D orbital of the hydrogen atom, you would need to multiply a lot of polynomials. Related polynomials play important roles in other problems in quantum mechanics and classic physics (Legendre Polynomials for example). 
Hopefully this clears some of it up, unfortunately in a large sense it is really a stepping stone, into the world of higher mathematics. They are important on their own, but for reasons that... also become much more obvious later on. 
A: Apart from the mundane reasons, e.g. "you need it to do certain computations" there are also some more technical reasons:


*

*The Stone-Weierstrass theorem: Let $X$ be a compact metric space and let $A$ be a $\mathbb{C}$-subalgebra of $C(X,\mathbb{C})$. If $A$ separates points in $X$ and vanishes at no point of $X$ and is self-conjugate, then $A$ is dense in $C(X,\mathbb{C})$. 
The polynomial functions on $[a,b]$ with complex coefficients are the prototypical example of such a subalgebra. Of course verifying that it is an algebra and has the stated properties is an application of multiplication of polynomials.

*Orthogonal polynomials. These are somewhat important in the theory of special functions. For example in [Andrews, Askey and Roy] there is an entire chapter devoted to families of orthogonal polynomials.
A: In digital signal processing a signal of finite length may be represented as a polynomial ( its Z transform).    So the convolution of two finite length signals is simply the multiplication of the Z transform polynomials.   
A: This is really several questions.  In increasing order of depth:


*

*Why multiply polynomials?  Because they are functions, and multiplication is a natural or desirable operation on functions.

*Why multiply functions?   Because the values of the functions are integers, or rational numbers, or real numbers, or objects of some other more complicated kind that can be multiplied (such as matrices or rotations).  If multiplication is natural for these numbers or number-like objects, it is natural for functions whose values are such objects.

*Why multiply integers, rational numbers, real numbers, etc?  Here is where explanation becomes difficult.  The natural operation is not multiplication but the more structured operation of "tensor multiplication" that retains information about the factors that are multiplied.  One can represent an Area as a product of type Length x Length that remembers its two-dimensionality, instead of a one-dimensional numerical value of that product that has forgotten its origins.  In the same way, for integer multiplication it is most direct to multiply 5 and 6 by drawing 5x6 as a rectangular 2-dimensional array of dots instead of the "numerical evaluation" of that array as a one-dimensional string of 30 dots. The latter is less natural in that it requires a method of enumerating the dots in the grid and there is no preferred order in which to count them.  This is also reflected in the ability to naturally multiply 5 gadgets by 6 gizmos, without there being any given ordering of the gadgets or gizmos.
Polynomials, especially polynomials in several variables ($x +3x^2y + 2 z^{10} x$ and such), have more inner structure than numbers and thus can reflect -- in fact, they can be defined and derived from -- the tensor structure of multiplication.   So it is not that polynomials exist and we might want to multiply them; it is that multiplication of numbers (or of finite sets) naturally involves more than just numerical information and polynomials are an enhancement of numbers that more directly embody that information so that the multiplication can be performed in a way that retains more of the inner structure.  
A hint of this is that integers in base 10 are values of polynomials at $x = 10$ (and those polynomials can be thought of as a "liberation" or "upgrading" of the numbers).  Multiplication of integers can sometimes be seen to replicate the patterns in the coefficients when the polynomials are multiplied for general $x$, e.g., compare powers of $102$ and $x^2 + 0x + 2$.  Later there are things like generating functions and convolutions that directly exploit polynomials as carriers of information that is enriched compared to using numbers alone.
(This is glossing over some technical questions about commutativity. Also, areas should be sums of LxL products, and one should explain the role of sums as well as products.  But these are details that do not affect the main point.)
In applications, multiplication represents interaction or correlation between different effects.  Situations where there are several independent processes isolated from each other and contributing to some outcome lead to sums  of functions of the different variables, such as $f(x) + g(y) + h(z)$, the terms in the sum accounting for the separate effects.  Situations where different parts of the mechanism can interact with each other in producing the outcome, or where there is correlation between different effects (e.g., one intensifies or suppresses the other), almost always involve multiplication when expressed mathematically.   For example, if you count the number of handshakes in a group of $n$ people the answer will be of order $n^2$, and if you count the number of possible 3-person interactions in this group it will be of order $n^3$.  Nonlinearities reflect the ability to organize the people into pairs, triples etc, and this again reflects the fact that the natural multiplication of two finite sets $A$ and $B$ is the set of ordered pairs $(a,b)$ of elements, one from each set, and the same for triples and higher numbers of sets. 
A somewhat faddish term for related ideas is (de- or re-)categorification.
A: When solving real-world problems with simple algebra, it is not uncommon to have polynomials scattered throughout the equation. Given that 1/(3x+4)=2x-1, which isn't an extraordinarily complex equation, the first thing you might want to do is multiply both sides by 3x+4, which means you will need to be able to deal with (2x-1)*(3x+4).
