ADDED: What I find unsatisfactory about my answer below is that I do not say who did what. I hope someone would post an answer that does this. I also have to say that although this classification of Thurston is (inevitably) more elaborate than for 2-manifolds, I find it quite beautiful.
Let me venture an answer, even though there are way, way more qualified people than me who will answer it correctly. I just want to see how close my understanding is to the correct answer. I will speak only about orientable 3-manifolds. I have no idea what happens for non-orientable ones.
First, if there are any 2-spheres in a compact 3-manifold that do not bound balls, then they can be made disjoint. By cutting along these spheres and plugging each hole with a ball, one gets a finite set of compact manifolds that are called "prime".
Next, one looks for incompressible torii (which I believe means that its fundamental group injects into the fundamental group of the manifold). These can also be made disjoint and the manifold chopped along these torii. At this point you don't try to fill in the toroidal hole.
Now everything reduces to a manifold without any incompressible spheres or torii, except it might have toroidal ends. There are three possible geometric structures for such manifolds:
- Spherical geometry
- Hyperbolic geometry
- Seifert fibrations (flat, solvable, nilpotent geometries)
What's notable about Seifert fibrations, as well as the 3-sphere, is that they are fibered by circles and admit a family of "collapsing" Riemann structures in the sense of Cheeger-Gromov-Fukaya. In other words, a family of Riemannian metrics where the sectional curvature remains bounded but the injectivity radius goes to $0$ everywhere. Thurston's conjecture implies that the only non-collapsible geometry is the hyperbolic geometry.
Before Perelman, proving the Thurston conjecture had been reduced to the following: Show that an orientable prime atoroidal 3-manifold, either compact or with toroidal ends that is not Seifert fibered or the 3-sphere has a hyerbolic structure.
Thurston had already proved that if these assumptions hold and the manifold is Haken (it has an incompressible surface of higher genus), then it is hyperbolic. Curt McMullen gave a beautiful proof of this via a proof of the theta conjecture.
ADDED: Note that this does not really answer the question. I don't believe that there is a complete classification of hyperbolic 3-manifolds (as there is for 2-manifolds). If that's right, then we don't really know what all 3-manifolds are.