I'm a little surprised no one has cited Thurston's impact on low-dimensional topology and geometry. I'm far from an expert, so I'm reluctant to say much about this. But I have the impression that Thurston revolutionized the whole enterprise by taking known results and expressing them from a completely new perspective that led naturally both new theorems and a lot of new conjectures. Perhaps Thurston himself or someone else could say something, preferably in a separate answer so I can delete mine.