Poincaré conjectured that three-dimensional shapes that share certain easy-to-check properties with spheres actually are spheres. What are these properties? My fellow geometer Christina Sormani describes the setup as follows:
The Poincaré Conjecture says, Hey, you've got this alien blob that can ooze its way out of the hold of any lasso you tie around it? Then that blob is just an out-of-shape ball. [Grigory] Perelman and [Columbia University's Richard] Hamilton proved this fact by heating the blob up, making it sing, stretching it like hot mozzarella, and chopping it into a million pieces. In short, the alien ain't no bagel you can swing around with a string through his hole.
That's zingier than anything the Times will run, but may still leave you without a clear picture of Perelman's theorem. Indeed, it's pretty hard to give an elementary account of the statement that Poincaré conjectured and that Perelman seems to have confirmed. (If that's what you're after, Sormani's home page links to a variety of expositions, including one in the form of a short story.) Instead, I'll try to explain why Perelman's theorem matters without explaining what it is.