The What? I'll let the Clay Mathematics Institute explain:
"Formulated in 1904 by the French mathematician Henri Poincaré, the conjecture is fundamental to achieving an understanding of three-dimensional shapes (compact manifolds). The simplest of these shapes is the three-dimensional sphere. It is contained in four-dimensional space, and is defined as the set of points at a fixed distance from a given point, just as the two-dimensional sphere (skin of an orange or surface of the earth) is defined as the set of points in three-dimensional space at a fixed distance from a given point (the center).
Since we cannot directly visualize objects in n-dimensional space, Poincaré asked whether there is a test for recognizing when a shape is the three-sphere by performing measurements and other operations inside the shape. The goal was to recognize all three-spheres even though they may be highly distorted. Poincaré found the right test (simple connectivity, see below). However, no one before Perelman was able to show that the test guaranteed that the given shape was in fact a three-sphere."
It's one of 7 "Millennium Problems" which has standing $1 million prizes for a solution at the Clay Mathematics Institute. Err, make that 6. Dr. Grigory Perelman in St. Petersburg has become the first winner of a Clay Millennium Prize. Only problem? He doesn't want it. He also failed to show at a 2006 Fields Medal ceremony from the International Mathematical Union.