the geometrization conjecture
The plan for surfaces is beautifully simple: every closed surface is a sphere, a torus, or a multi-holed surface, and each gets exactly one uniform geometry. Thurston proposed that three-manifolds obey an analogous, only slightly more elaborate, plan: cut any closed 3-manifold along a canonical collection of spheres and tori, and each resulting piece carries one of the eight model geometries. Geometrization is this organizing dream, now a theorem, that tames the wild zoo of 3-manifolds into geometric Lego.
Precisely: every closed orientable 3-manifold can be cut, first along a finite collection of disjoint embedded 2-spheres into prime pieces (the connected-sum or Kneser-Milnor decomposition), and then each prime piece cut along a canonical minimal collection of disjoint incompressible tori (the JSJ decomposition), so that every remaining piece is either Seifert-fibered or has a complete finite-volume geometric structure modeled on one of the eight Thurston geometries. Equivalently: the interior of each piece admits a homogeneous Riemannian metric locally isometric to one of S^3, E^3, H^3, S^2 x R, H^2 x R, SL(2,R)-tilde, Nil, or Sol. The hyperbolic geometry H^3 is the generic and richest case.
It was conjectured by Thurston around 1980 (he proved the case of Haken manifolds) and proved in full by Grigori Perelman in 2002-03 using Hamilton's Ricci flow with surgery. Geometrization contains the Poincaré conjecture as a special case: a simply connected closed 3-manifold geometrizes as a quotient of S^3 with trivial group, hence is S^3 itself. The honest caveat is that the proof does not 'just smooth everything out' — Ricci flow on a 3-manifold develops singularities (necks pinching off), and Perelman's surgery, performed infinitely often if needed, is exactly what cuts along the geometric decomposition.
Take the connected sum (T^3) # (figure-eight knot complement filled to a closed manifold). Geometrization first splits the connected sum along a 2-sphere into the two prime summands. The T^3 piece is geometric of type E^3; the once-hyperbolic summand is geometric of type H^3. No single homogeneous metric works on the whole manifold — the sphere decomposition is forced, and the two pieces wear different geometries.
Geometrization in action: a connected sum splits along a sphere into an E^3 piece and an H^3 piece.
Geometrization is a theorem (Perelman), not a conjecture, since 2003 — but it does NOT say every 3-manifold is geometric; it says every 3-manifold canonically decomposes into geometric pieces. The Ricci flow proof requires surgery precisely because the flow alone forms singularities and cannot be run forever unaided.