The Poincare Conjecture, 1900:
If M is a 3-manifold with trivial fundamental group, and Π_i(M)=0 for i=1,2 and =Z for i=0,3 (ie, M has the homotopy groups of a 3-sphere), then M is homeomorphic to the 3-sphere.
Simply: (1904) That if any loop on the surface of a three dimensional shape can be shrunk to a point (as any loop can on a 3-D sphere) then the shape is just a 3-D
Originator: Jules Henri Poincaré, 1854-1912.
Incentive: $US1million, one of the seven Millennium Prize Problems of the Clay Mathematics Institute.
Notable false proof: JHC Whitehead, 1934.
Has led to: Interesting new examples of 3-manifolds; several celebrated cases of Poincaritis.
Unusual aspect: Solving this problem in four and more dimensions has been much easier than solving it in three.
Likely proof: Grigori Perelman, Steklov Institute of Mathematics, St Petersberg, 2002 and 2003, although the Clay prize has yet to be awarded.
Link to a nice YouTube: http://youtu.be/9sfkw8IWkl0