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Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

Categories abstract the essential properties of sets and functions. They capture the importance of compositionality and showcase the overreaching breadth of associative operators. What are they? They are nothing more than monoids ---sets with an associative operation that has a unit-- but the operation of which is typed. A key example is matrix multiplication, it yields a monoid when restricted to square matrices, but generally is a category! Of course this is only the tip of the iceberg!

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