Lots of examples.
A recent one would be the use of gauge theory in topology. Sieberg-Witten theory provides us a means of producing many different smooth structures on certain 4-manifolds (for example, by certain knot surgeries on the elliptic fibration E^2, which we have been discussing in our topology seminar here--this changes the Sieberg-Witten invariants without changing the topology, so it yields examples of topologically, but not smoothly equivalent manifolds). Many topologists might take it as a black-box, but it's based on such things are spinors, Dirac operators, gauge theory. Sieberg-Witten theory also appears to have been a big influence on such things as Heegard-Floer homology, which is a hot topic in topology, now. And of course, before Sieber-Witten, there was Donaldson theory, which might be thought of as a more primitive version of it. This was inspired by Yang-Mills gauge theory.
A related thing, which is what I study, is topological quantum field theory. It's very bare-bones quantum mechanics, I would say. It has some significance in pure math, but I think the interest in it comes more from physical considerations.
Another related thing is the Jones polynomial, which is a knot invariant. Jones was inspired here, originally, by studying Von Neumann algebras, but there were various connections to physics that were revealed later. The first math paper I ever read showed how the Jones polynomial could be thought of as some kind of partition function of a knot diagram. Also, Witten gave it an interpretation in terms of Chern-Simons theory and path-integrals.
Another closely related example would be the theory of quantum groups, which began as an effort (by physicists) to produce solutions of the Yang-Baxter equation from physics.
These are all pretty recent developments (last 30-40 years).
Another example would be the use of mirror-symmetry from string theory to solve certain enumeration problems in algebraic geometry.
The general theory of relativity influenced the development of modern differential geometry.
