Typo: a "closed manifold" is compact
without boundary.
The Poincaré conjecture states that closed simply connected 3-manifolds are all diffeomorphic to the 3-sphere. Even stronger, every closed 3-manifold whose fundamental group is finite is a quotient of the 3-sphere by a discrete subgroup of its group of isometries, SO(4) (called the elliptization theorem, which is what Perelman proved). I've been told some astronomers once looked into whether the cosmic background radiation suggested that we lived in Poincaré dodecahedral space.
Thurston's geometrization conjecture (all proved as of 2012) is that all closed 3-manifolds can be built out of certain 3-manifolds with standard Riemannian geometries by gluing them together along their torus boundaries and by introducing wormholes, essentially. Some interesting cases are the spherical geometries (constant positive curvature, classified above) and hyperbolic geometries (constant negative curvature, also classified by Perelman). The only actually flat closed 3-manifolds are the 10 finite-order mapping tori of the torus -- one example is S^1 x S^1 x S^1, where the 3-dimensional version of the game Asteroids would be played.
There are infinitely many closed hyperbolic 3-manifolds. I don't understand why space can't be negatively curved.
Jeff Weeks has a cool program for flying through different spaces: http://geometrygames.org/CurvedSpaces/index.html
4-manifolds are definitely more complicated. Lots of techniques that work for 3-manifolds and (5+)-manifolds don't work.