A branch of mathematics I want to study one day.
an extreme form of

algebra.

In category theory, the primitive notions are nodes and arrows. A category is a set of nodes connected with arrows such that if a path leads from one node to another, so does a direct arrow; in other words, a transitive directed graph.

Nodes represent sets; arrows represent functions; categories describe particular types of sets entirely in terms of the functions that operate on them, and on related sets.

This avoids the overspecification you often get when describing mathematical objects in set theoretic terms.
Take a look at the definition of tuples in terms of sets, for instance: < x, y, z > is defined as { x, {y, {z}}}. Clearly this construct 'behaves like' a tuple but it cannot really be said to be 'the real definition' of a tuple. Category theory tries to do away with this nonsense.

That's as far as I understand it. Remind me to add to this when I finish the textbook.