Is there a complete classification of constant mean curvature surfaces?

Karsten Grosse-Brauckmann, Rob Kusner, and John Sullivan have written on the classification of embedded CMC surfaces for quite some time. I think a reasonable place to read about this program is this survey by Rob Kusner. You might also want to look at some of their other papers on the arxiv as well.

A nice idea used in their work which goes back to a paper of Korevaar, Kusner and Solomon in the 80's is that embdedded CMC surfaces have "ends" which asymptote to Delaunay surfaces - furthermore these ends can be assigned a "tension" (this should be familiar to you as the parameter which distinguishes between Delaunay surfaces) and these tensions satisfy a force balance rule.

I think the story for minimal surfaces is much better understood due to the work of Meeks and Minicozzi though I haven't read their work in much detail.

Since my previous answer to this question major progress was made: As Robert Haslhofer already mentioned, Brendle proved the Lawson conjecture that the only minimal CMC torus in the 3-sphere is the Clifford torus. Building on this work, Andrews and Li ("Embedded constant mean curvature tori in the three-sphere." J. Differential Geom. 99 (2015)) classified all embedded CMC tori in $S^3.$ To be more concrete, they used the same two point function as in Brendles proof in order to show that every embedded CMC torus must be rotationally symmetric, and hence is classified and can be written down in terms of elliptic functions explicitly.

The case of higher genus CMC surfaces in space forms is more difficult and unsolved by now. But there has been done computer experiments which suggest that the space of embedded CMC surfaces of higher genus $g\geq2$ with certain symmetries is related to the space of embedded CMC tori, see http://arxiv.org/pdf/1503.07838.pdf and the following image:

Let me add some uniqueness theorems for CMC and minimal surfaces:

1) A classical theorem of Hopf says that any immersed CMC sphere in $\mathbb{R}^3$ is the round sphere.

2) A classical theorem of Aleksandrov says that any embedded closed hypersurface in Euclidean space with constant mean curvature is the round sphere.

3) Very recently, Simon Brendle proved the Lawson conjecture: Any embedded minimal torus in $S^3$ is congruent to the Clifford torus, see arXiv:1203.6597v2.