Derivatives seem easy to understand abstractly as the rate of change of something, higher order derivatives are the rate of change of the rate of change of something, and so on.
I, however, have trouble understanding what an integral is in a general sense. It can be thought of as the sum of the infinitely small rectangles in an shape, but what is it, with respect to the initial function? I don't really understand why integration is the inverse operation of derivation. either.