In quantum theory one thus learns to like noncommutative, but still associative, algebras. It is interesting however to note why associativity without commutativity is studied so much more than commutativity without associativity. Basically, because most of our examples of binary operations can be interpreted as composition of functions. If we try to generalize the heck out of the concept of a group, keeping associativity as a sacred property, we get the notion of a category.
Categories are some of the most basic structures in mathematics.
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.