They were created by Samuel Eilenberg and Saunders MacLane. I invented it to talk about natural transformations. Many nice things in mathematics are functors. A functor is a kind of map between categories. That is, objects go to objects and morphisms go to morphisms.
Category theory is popular among algebraic topologists.
Typically an algebraic topologist will try to assign algebraic invariants to topological structures. For example, the fundamental group is functorial. Topologists know how to cook up a group called the fundamental group from any space. But ALSO, any map between spaces determines a homomorphism of the fundamental groups. So the fundamental group is really a functor from the category Top to the category Group.