You might like to read Introduction to Integration first!
Integration
Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area under the graph of a function like this:
 
The area can be found by adding slices that approach zero in width:
And there are Rules of Integration that help us get the answer.

Notation
The symbol for "Integral" is a stylish "S"
(for "Sum", the idea of summing slices): 
After the Integral Symbol we put the function we want to find the integral of (called the Integrand),
and then finish with dx to mean the slices go in the x direction (and approach zero in width).
Definite Integral
A Definite Integral has start and end values: in other words there is an interval (a to b).
The values are put at the bottom and top of the "S", like this:
Indefinite Integral (no specific values)  Definite Integral (from a to b) 
We can find the Definite Integral by calculating the Indefinite Integral at points a and b, then subtracting:
Let's try another example:
And another example to make an important point:
But we can have negative areas, when the curve is below the axis:
Try integrating cos(x) with different start and end values to see for yourself how positives and negatives work.
But sometimes you want the actual area (without the part below being subtracted):
Continuous
Oh yes, the function we are integrating must be Continuous between a and b: no holes, jumps or vertical asymptotes (where the function heads up/down towards infinity).
Properties
Reversing the interval
Reversing the direction of the interval gives the negative of the original direction.
Interval of zero length
When the interval starts and ends at the same place, the result is zero:
Adding intervals
We can also add two adjacent intervals together:
Summary
The Definite Integral between a and b is the Indefinite Integral at b minus the Indefinite Integral at a.