Is there some general description of all homology 3spheres?
Certainly. There's a general description of all compact 3manifolds now that geometrization is about. So for homology 3sphere's you have the essentially unique connect sum decomposition into primes. A prime homology 3sphere has unique splice decomposition (Larry Siebenmann's terminology). The splice decomposition is just a convienient way of encoding the JSJdecomposition. The tori of the JSJdecomposition cut the manifold into components that are atoroidal, so you form a graph corresponding to these components (as vertices) and the tori as edges. The splice decomposition you can think of as tree where the vertices are decorated by pairs (M,L) where M is a homology 3sphere and L is a link in M such that M \ L is an atoroidal manifold. By geometrization there's not many candidates for pairs (M,L). The seifertfibred homology spheres that come up this way are the Brieskorn spheres, in that case L will be a collection of fibres in the Seifert fibering. Or the pair (M,L) could be a hyperbolic link in a homology sphere. That's a pretty big class of manifolds for which there aren't quite as compact a description, compared to, say, Brieskorn spheres. 


On the other hand, there is no particular classification of hyperbolic homology 3spheres, much less hyperbolic links in homology 3spheres, other than in general terms that they all come from hyperbolic groups. For instance, part of geometrization establishes that if a finite group acts freely on $S^3$, then it is equivalent to an action by isometries on a round $S^3$ and is a subgroup of $\mathrm{SO}(4)$. Before geometrization, Milnor and Lee established severe restrictions on how a finite group $G$ can act freely on any homology 3sphere, with the case of $S^3$ particularly in mind. Either $G$ is a spherical group, or it is one other family that hasn't been excluded. For all we know, if $G$ acts freely on any homology 3sphere, then it acts on $S^3$ too. I think that this is still an open problem, and geometrization by itself doesn't settle it. The working description homology 3spheres for many purposes, in particular quantum topological invariants, is rather different. In practice, a homology 3sphere is often given by surgery on a link in $S^3$ (or in some other homology 3sphere) whose matrix has determinant 1. The big drawback of course is that the description is far from unique. 


A nice historical note  Dehn observed that if M and N are knot complements and if you glue M to N switching meridian and longitude then the result is a homology sphere. Of course this is a special case of what Ryan was saying. Another nice fact: the Poincare homology sphere is the only one with finite fundamental group. 


Another way to represent homology spheres is to take a Heegaard splitting for $S^3$, cut and reglue by an element of the Torelli group. This is not canonical, but any two Heegaard splittings are equivalent after some number of stabilizations. If you wanted to enumerate every homology sphere, you could list elements of the Torelli group, and construct 3manifolds, then throw away repeats by using some solution to the homeomorphism problem for 3manifolds. This is not really feasible to carry out in practice, but is one way to give a "general description" of homology spheres at least in theory, by giving a recursive enumeration of them. 

