So what is the Poincaré Conjecture?

Wolfram's site explains it quite well.

In its original form, the Poincaré conjecture stated that every simply connected closed three-manifold is homeomorphic to the three-sphere. More colloquially, the conjecture says that the three-sphere is the only type of bounded three-dimensional space possible that contains no holes. This conjecture was first proposed in 1904 by Poincaré.

Soon thereafter this specialized conjecture for the case n=3 was generalized to the conjecture that every compact n-manifold is homotopy-equivalent to the n-sphere if and only if it is homeomorphic to the n-sphere, which is logically equivalent to the earlier weaker conjecture in the case n=3. The n=1 case is trivial, the n=2 case was known to 19th century mathematicians, n=4 was proved by Freedman in 1982 for which he was awarded the 1986 Fields medal, n=5 was proved by Zeeman in 1961, n=6 was proved by Stallings in 1962, and n>6 was proved by Smale in 1961 (although Smale subsequently extended his proof to include all n>4).

The n=3 case has baffled mathematicians since it was first proposed, spawning a multitude of incorrect proofs by many mathematicians, including Poincare and Whitehead. However, it now appears that this remaining case has finally been proved by Perelman although the proof has not yet been fully verified. In fact, Perelman's work may prove the more general Thurston Geometrization Conjecture from which the Poincare Conjecture would then follow.

Perelman's two papers are The entropy formula for the Ricci flow and its geometric applications and Ricci flow with surgery on three-manifolds. For the non Fields Medal candidates among us, a more accessible (but still highly technical) discussion can be found on the AMS site.