"it is an open question if the known compact manifolds in 3-D are complete."
This is a quote from Eric Weisstein's CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480. (Google Books link)
Is this still the case, post-Perelman? What are the known compact manifolds in $\mathbb{R}^3$?
I ask this as someone (obviously!) naive in these areas. Thanks for educating me!
Actually, it's very easy to write down a complete list of orientable 3-manifolds (and thus all, since every nonorienatble 3-manifold has its orientation cover), using a tool called Heegaard splitting. This list has the following structure: break it up into $\mathbb{N}$ many sublists corresponding to different genera of 2-dimensional manifolds. Having fixed $g$, look at all mapping classes from a surface with that genus to itself (these form a finitely generated group), and then consider the 3-manifold obtained by gluing two handlebodies (filled genus $g$ surfaces) using that map to identify the boundaries.
The proof that this is a complete list of 3-manifolds is quite easy; it can be explained in a few minutes to a smart undergrad. The hard part isn't listing all 3-manifolds; it's that the list I gave above (and various other lists one can produce with similar techniques, such as surgery) is massively redundant. We even know the rules (analogues of Reidemeister moves) that tell you all the redundancies, but just as Reidemeister moves don't help that much with classifying knots, this doesn't mean you can take two Heegaard splittings and efficiently test if they give you the same 3-manifold.
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:
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.
I like to think that compact 3-manifolds will not be known until we know "the list" of compact, oriented, hyperbolic 3-manifolds, in the way that we know "the list" of compact, oriented, hyperbolic surfaces: the genus 2 surface, the genus 3 surface, the genus 4 surface, ...
We don't know "the list" yet, as the other answers indicate. But the potential for ordering hyperbolic 3-manifolds by volume was established by Jorgensen who proved that the set of volumes of finite volume hyperbolic 3-manifolds is a well-ordered set with order type $\omega^\omega$ (the same order type as polynomials with natural number coefficients), and each volume occurs for only finitely many manifolds. And with that in mind, we know the first entry: the Weeks manifold, which is the unique lowest volume closed hyperbolic 3-manifold, as proved by Gabai, Meyerhoff, and Milley. I'm not up on the very latest developments of this technology, but I think that the next few lowest volumes have also been completely listed, using the MOM technology of the same three authors.
