Short answers, pulled from the story.
The Poincaré conjecture states that every three-dimensional topological manifold that is closed, connected, and has a trivial fundamental group is homeomorphic to the three-dimensional sphere. Henri Poincaré posed the question in 1904, and it remained unproven for nearly a century before Grigori Perelman published his proof in 2002-2003.
Grigori Perelman proved the Poincaré conjecture by posting three papers to the arXiv preprint server beginning on the 11th of November 2002. His proof built on and completed a program developed by Richard S. Hamilton using a technique called Ricci flow with surgery.
Perelman declined the Clay Mathematics Institute's one-million-dollar Millennium Prize, awarded on the 18th of March 2010, stating that Hamilton's contribution to the proof had been equal to his own. He also declined the Fields Medal in 2006.
Ricci flow is a set of equations introduced by Richard S. Hamilton in 1982 that deforms the metric of a manifold over time, analogous to the way heat diffuses through a solid. Perelman extended Hamilton's program by characterizing every singularity Ricci flow can produce and introducing Ricci flow with surgery, which cuts the manifold at singularities and allows the flow to continue on the resulting pieces until only spheres remain.
Perelman was awarded the Fields Medal in 2006 and the Clay Millennium Prize in 2010, worth fifteen thousand Canadian dollars and one million US dollars respectively; he declined both. Hamilton received the Leroy P. Steele Prize for Seminal Contribution to Research in 2009 and the Shaw Prize in 2011.
The mathematical community recognized the proof in 2006, after three independent groups of mathematicians verified Perelman's papers between May and July of that year. The journal Science named it the Breakthrough of the Year in December 2006, and John Morgan declared at the International Congress of Mathematicians on the 24th of August 2006 that Perelman had solved the conjecture in 2003.