TITLE: What is the difference between integrals and contour integrals?
QUESTION [5 upvotes]: I understand integrals but what are contour integrals?

REPLY [3 votes]: Integrals, as you call them, and contour integrals are the same thing at a basic level: they are limits of certain sums over functions.  The practical difference, however, is that the contour integral really represents one way, out of infinitely many, to integrate a function between 2 points, Point A and Point B, in a plane.  The integral, on the other hand, has only one path by which to integrate from Point A to Point B.  
Further, it makes sense to speak of closed contour integrals - well, OK, you could speak of integrating a function from Point A to Point B, and back again on the real line, but that isn't very interesting.  On the other hand, you can speak of going from Point A to Point B along one path, and then from Point B back to Point A along a different path.  In some cases, the net result is not zero.  (In fact, it only is when the function in question is analytic on and within the contours.)