Category theory and in particular Yoneda's lemma. Thinking of mathematical objects as part of a category changes eveything. It allows one to define an object only by the way it relates to other objects (universal problems, Yoneda's lemma).

Obviously people knew that $K[X]$ was a the free $K$-algebra on one element long before category theory was invented but expressing it in the abstract language of categories leads to a much deeper understanding and much much simpler proofs. Just think of how painful it would be to study tensor products of modules without their functorial characterization for example.

And the mathematical community owes a great debt to Grothendieck for showing how powerful this point of view can be.