A commutative diagram is just a collection of nodes and directed edges between nodes (aka, a directed graph), but it's a directed graph along with the claim that any two paths in this graph which start at the same node and end at the same node are to be considered equivalent, in some sense.

In general, a directed (multi)graph along with an account of which of its paths are and are not to be considered equivalent to each other (where this equivalence relation satisfies some basic nice properties) is known as a "category". This concept comes up ubiquitously in math/abstract logic/etc. Commutative diagrams are useful for quickly visually reasoning about equivalences of paths in such contexts.