The "Yamabe problem": Every compact Riemannian manifold admits a conformally-related metric with constant scalar curvature. Yamabe thought he had proved this in 1960, but his proof had--I'm not making this up--a sign error. The error was discovered by Neil Trudinger in 1968, after Yamabe's death. As I understand it, Trudinger was working on a similar nonlinear elliptic PDE problem (with a critical Sobolev exponent) and got stuck, so he looked at Yamabe's paper to see how Yamabe had dealt with the same issue. Turned out he hadn't. Trudinger was able to give a partial solution to the problem; later Aubin expanded it to cover more cases, and finally in 1984 Rick Schoen was able to prove it the full theorem cases that Aubin had left open (with a small gap in the higher-dimensional case that was repaired by Schoen and Yau in 1988). The proof surprisingly used the positive mass theorem from general relativity.
Yamabe's original paper never attracted much attention until the error was found. But because of the subtlety of the methods required to fill in the gap, it has become a model for applications of nonlinear elliptic PDE to geometry, especially to conformally invariant problems and other problems with critical regularity.
