I would like to supplement others' very nice answers with this beautifully written article by Hamilton on the Nash-Moser theorem. It describes several nice applications of the theorem at the end, including:

• Existence of embeddings of surfaces with curvature bounds
• Existence theorems for the shallow water equations
• If $S$ is a submanifold of a Riemannian manifold $X$ then the space of submanifolds near $S$ with the same volume is a codimension $1$ submanifold of the space of submanifolds
• Nearby symplectic/contact structures can be conjugated via a diffeomorphism (up to cohomological restrictions)
• The diffeomorphism group of a compact manifold is a principal bundle over the space of smooth probability measures

I'm not sure if Nash himself actually proved any of these results, but I would include them as part of his legacy. It would seem that the Nash-Moser theorem (and its generalization, the h-principle, as someone else mentioned) is the larger theory into which Nash's ideas have been absorbed.

Interestingly, Nash's original embedding theorem for Riemannian manifolds was not presented as an application of the Nash-Moser theorem. Does anyone know if the former actually follows from the latter, or do they just have similar proofs?