What is an integral and the anatomy thereof? I am currently in differential Calculus.  Derivates resonate well with me.  However, half of the questions asked on here relate to Integration. 
This question is as general as it appears.  I'm sure it will serve well to myself and other members like me.
What is an Integral? What do the numbers indicate on the $\int$?
Thank you.
 A: It's certainly too general a question to give a complete answer here. I'll give a few elements.


*

*the integral $\int_a^b f(x) dx$ can be seen as the area under the graph of the function $f$ between $x=a$ and $x=b$. The symbol $\int$ historically represents a sum, as integration can be thought of as a limit of sums of the form $\sum_k \frac{b-a}{n} f(a+ k \frac{b-a}{n}).$

*Integration is the inverse procedure to derivation : the derivative of the primitive of $f$ is $f$, where the primitive of $f$ is a function $F$ defined (up to the choice of the constant $a$) by $F(x)=\int_a^x f(t)dt$.

*the quantity $\frac{1}{b-a} \int_a^b f(x)dx$ represents the mean of the function $f$ between $a$ and $b$. In probability theory, if $f=X$ is a random variable, it represents the expectancy of $X$. 
A: Well, I'm assuming you're dealing with single variable calculus, i.e. dealing with functions from the real line to the real line. If that's the case, the integral defines one kind of sum which is motivated with the problem of finding areas. I'll give you the motivation and the main elements of the definition from Spivak's Calculus. I won't prove the necessary lemmas that allows us to show that the definition is a good definition, but I really recommend you look after Spivak's book. What I'll define is the so called "Darboux integral". Many people prefere to start introducing integrals by the "Riemann integral", however Darboux integral seems more intuitive for me. Also you can show that any Riemann integrable function is Darboux integrable. Let's get started:
Let's imagine we have a function $f : I \subset \mathbb{R} \to \mathbb{R}$ and let's suppose that $\left[a, b\right]\subset I$ and that in this subset $f$ is bounded. There's a region in the $xy$ plane that is bounded by the horizontal lines $x = a$ and $x =b$, and by the axis $y=0$ and the graph of $f$. Imagine you want to define precisely what is the area of this region.
Well, your intuition allows you to imagine what should be the area of a rectangle: lenght of it's basis times it's height. So, what we do is that we consider what we call a partition $P$ of $[a,b]$. This is a finite set of points $P = \left\{t_0, t_1\dots t_n\right\}$ such that we have $a=t_0 <\cdots < t_n = b$. Intuitively, this collection of points allows one to chop the interval $[a, b]$ into smaller intervals $I_k=[t_{k-1}, t_k]$.
Our objective is to approximate the region by rectangles and define the area as a sum of those things we would call the areas of the rectangles intuitively. The intervals defined already allows us to talk about the basis of our rectangles, but what about the heights? Well, we do as follows, we define $M_k = \sup\left\{f(x) \mid x \in I_k\right\}$ and $m_k =\inf \left\{f(x) \mid x \in I_k\right\}$.
Hence, if we take the rectangles with basis $I_k$ and height $M_k$ we'll get a region that approximates $f$ "from above" and if we take the heights $m_k$ we'll get a region that approximates $f$ "from bellow". We then define what we'll call the lower and upper sums of $f$ in the selected partition.
The lower sum of $f$ in the partition $P$ is defined as:
$$L(f, P)=\sum_{k=1}^{n}m_k (t_k-t_{k-1})$$
And the upper sum of $f$ in the partition $P$ is defined as:
$$U(f, P) = \sum_{k=1}^{n}M_k(t_k - t_{k-1})$$
Our intuition tells us that $U(f, P)$ approximates the area from above while $L(f, P)$ approximates the area from bellow. We then show that if we refine the partition, in other words, if we add points to the partition then the lower sum never decreases and the upper sum never increases. In other words, we show that it is possible that those sums converge to a common value after refining enough. Our intuition says that this common value must be the area we seek to define. So we simply take the set of all upper sums, and the set of all lower sums. If the the values converge to a common value the infimum of the set of upper sums will equal the supremum of the set of lower sums (use your intuition), and then we make the defintion:
Let $f: I \subset \mathbb{R} \to \mathbb{R}$ bounded on $[a, b] \subset I$. If we have:
$$\sup \left\{L(f, P) \mid P \ \text{is partition of} \ [a,b]\right\} = \inf \left\{U(f, P)\mid P \ \text{is partition of} \ [a,b]\right\}$$
Then we say that $f$ is integrable on $[a,b]$ and we call this common value the integral of $f$ on $[a,b]$ which we denote $\int_{a}^{b}f$
Now look what you've done: you caught your intuition about area, and used it just to guide you finding a precise definition for the area of certain regions. Although guided by intuition, this definition is precise and general. If you follow the thought you'll get what the integral is on the real line. The relation to primitives is stated by the fundamental theorem of calculus and I shall not make it precise here. Look at Spivak, you'll learn a lot. I hope this helps you somehow.
A: The integral of a function is it's antiderivative. So for example $\int x^2 dx= \frac{x^3}{3}+c$ and when we differentiate $\frac{x^3}{3}+c$ we get $x^2$ back. When you have numbers on the top and bottom of the integral sign then the integral is called a definite integral. In this case it is interpreted as the area between the function being integrated and the x-axis between the two given numbers. So for example $\int \limits_0^1 x^2 dx$ is the area between the x-axis and the parabola $x^2$ between the values 0 and 1 on the x-axis.
A: There are many different types of integrals, but I guess you just started of with calculus and thus mean Riemann integrals over functions from $D\subset\mathbb R$ to $\mathbb R$.
In this case, the idea is to determine the area limited by the function graph and the horizontal coordinate axis. If you want to calculate $\int_a^bf(x)dx$ for example, you have to find an antiderivative of $f$ first, i.e. a function $F$ with $F'=f$. Then $\int_a^bf(x)dx=F(b)-F(a)$.
A concrete example: Let $f:\mathbb R\rightarrow\mathbb R,x\mapsto f(x)=x^2$. We want to determine the area limited by the graph of $f$ and the horizontal axis between $x=0$ and $x=2$. An antiderivative of $f$ would be $F(x)=\frac 1 3x^3$, because $F'(x)=x^2=f(x)$. Now we can calculate $\displaystyle\int_0^2f(x)dx=F(2)-F(0)=\frac 1 3\cdot2^3-\frac 1 3\cdot 0^3=\frac 8 3$.
A: When no numbers are on the integrand (i.e. $\int$ vs. $\int_a^b$), the integral is called indefinite.  It is, essentially, an antiderivative:
$$F'(x) = f(x) \implies \int f(x)\,dx=F(x)+C$$
where $C$ is an arbitrary constant.  The $dx$, much like in differentiation, represents the variable which is.  This is equivalent to saying 
$$ \frac{d}{dx} \int f(x)\,dx = f(x)$$
If there are numbers along the integrand, this integral is called definite:
$$\int_a^b f(x)\,dx = \text{Area}$$
and you can think of this as the area between the function being integrated and the $x$ axis, over $(a,b)$.  This concept is closely related to antidifferentiation, hence the same symbol is used.
