Abstract algebra is often about studying structure on sets. That usually takes the form of studying operator(s) with certain properties and the elements that they work with. Some early examples that students see are things like rotations/reflections of n-gons, modular arithmetic, polynomials, matrix spaces, and fields.

It is different from regular algebra in the sense that often one starts with a set and operation one supposes to have a particular kind of property (perhaps it's a group, or perhaps it's a field, or perhaps it's a certain kind of field (maybe separable, or local, or ...)). Then from these assumptions one proves facts about the assumed algebraic structure.

It's different in that often particular elements aren't of much interest, and it's not as tied down to a particular context.