The way to carry out a double integration is to integrate f(x) first with respect to x. Then, the inside integral sign will have limits of integration for variable x. They go in for x.
Then take this function (once evaluated at the limits of integration for x) and integrate the expression with respect to variable y, and lastly evaluate it at it's limits.
Just as in the single variable case the double definite integral give you a number, and a good check is if you get a function (containing variables) you probably messed up the order of integration. Remember however that the variables can be switched to suit the problem if it is difficult to start with a certain integration.
So the basic concept is to remember to do the inside integral first, evaluate it, then do the outside. They can be switched to suit the difficulty of the problem.
For a more geometric meaning of the double integral I won't type it some one might, but this is basically how you evaluate them.