[vouaobrasil]

Set theory is basically a study of the formation of single sets of objects and the study of various infinities. Typically mathematical objects are sets with operations on them so set theory is useful for understanding them, but the vast majority of set theory is not necessary for the rest of mathematics.

Category theory is the study of collections of mathematical of a give type. The category of groups, the category of sets, the category of vector spaces. The key facet of category theory is that you can have "functions" (called functors) between categories and the power of category theory is the study of these functors. I put "functions" in parentheses because most categories are not sets in the set-theoretic sense because they do not have a well-defined cardinality. Of course, some categories called "small categories" are sets.