The usual intuitive understanding of integration is the area under a function, or more generally, an accumulation of something. For example, here's a graphical representation of an integral:
$$ s = \int_a^b f(x) dx $$
It's very important to understand why the integral is some sort of 'opposite' of the derivative (this is, of course, the fundamental theorem of calculus). Try this: as you said, a derivative is a rate of change. So if we consider the graphical interpretation of the integral shown above, what is the rate of change of the area? Well, the higher the function is, the faster the area under it increases, so the rate of change is just the value of the function. In terms of calculus, this means the derivative of the integral is the original function.
Conversely, what is the integral of the derivative? The derivative is the rate of change, and the integral is an accumulation. So if we accumulate, or collect, all of the tiny changes as the function goes along; well, that's also just the original function.
I hope these explanations help, but keep in mind that as you do more work with these concepts and learn more results that make use of them, you'll develop your own intuitive understanding of what they mean. And that understanding will be much more valuable than anything someone else can explain to you.