Category theory is a way of taking theorems in one area (say, group theory), figuring out the minimum set of assumptions necessary for the theorem to hold, and

Category theory is a way of taking theorems in one area (say, group theory), figuring out the minimum set of assumptions necessary for the theorem to hold, and then translating it into every possible similar structure. It's sort of like interfaces in some ways. Because normal mathematics is already pretty darned abstract, category theory gets even worse, but there is some content there underneath it.