Every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.
Loosely speaking, this means that if a given "three-dimensional object" has a set of sphere-like properties (most notably that all loops in it can be shrunken to points), then it is really just a "deformed version" of a 3-sphere.
The conjecture has induced a long list of false proofs, and some of them have led to a better understanding of low-dimensional topology.
Analogues of the Poincaré conjecture in dimensions other than 3 can also be formulated:
Every compact n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.
The Poincaré conjecture as given above is equivalent to the case n=3. The difficulty of low-dimensional topology is highlighted by the fact that these analogues have now all been proven (with dimension n=4 being the hardest one by far), while the original 3-dimensional version of Poincaré's conjecture remains unsolved.
Its solution is related to the problem of classifying 3-manifolds. A classification of 3-manifolds is generally accepted to mean that one can generate a list of all 3-manifolds up to homeomorphism with no repetitions. Such a classification is equivalent to a recognition algorithm, which would be able to check if two 3-manifolds were homeomorphic or not.
One can regard the Poincaré Conjecture as a special case of Thurston's 25-year-old Geometrization Conjecture. The latter conjecture, if proven, would finish off the quest for a classification of 3-manifolds. The only parts of the Geometrization Conjecture left to be proven are called the Hyperbolization Conjecture and the Elliptization Conjecture.
The Elliptization Conjecture states that every closed 3-manifold with finite fundamental group has a spherical geometry, i.e. it is covered by the 3-sphere. The Poincaré Conjecture is exactly the subcase when the fundamental group is trivial.
In late 2002, reports surfaced that Grigori Perelman of Steklov Mathematical Institute, Saint Petersburg might have found a proof of the geometrization conjecture, carrying out a program outlined earlier by Richard Hamilton. In 2003, he posted a second preprint and gave a series of lectures in the United States. His proof is still being checked.