(I agree with others that this is a near-duplicate, and will transfer my answer there if needed.)
The power of naming...
J.-M. Souriau wrote a book meant to take anyone from the crib to relativistic cosmology and quantum statistics — that he unfortunately (predictably?) never got to really finish, but the draft is online. He starts by saying that all babies are (of course) natural born mathematicians, and everyone’s “first and sensual” abstraction is when they name that warm pink blob “mommy”.
Surely that qualifies as “crucial to further understanding”; he argues that half math is more of the same, and tells stories illustrating the power of naming numbers, points, the unknown $(x)$, maps, variables, determinants, groups, morphisms, cohomologies... all would qualify as answers. It’s not just about coming up with epoch-making definitions, either: teaching students to define and name variables in their problems is also what we do.
We’ve all seen quotes to the effect that it’s all in the definitions. “Success stories” in that are largely the history of math, or at least one way to look at it. Specific recent ones that likely inspired the book (and made it to the MSC) would be symplectic manifold (1950), symplectic geometry (1953; earlier it was this), and moment(um) map (1967).