Curvature of a finite metric space

It is not a problem to define, Alexandrov's comparison inequalities make sense for all metric spaces. Check (3+1) and (2+2) point comparison in our book. However, it is not clear what to do with these spaces (I do not know anything interesting about them unless they have length metric.)

A different definition (defining bigger class of spaces) is given by Nikolaev and Berg, see this paper and the references there in.

The question which spaces admit a distance preserving map into CATs and CBBs is open, it is discussed here in section 7.

If something more analogous to Ricci curvature than to sectional curvature would interest you, then there has been some work done on the Ricci curvature of discrete spaces. I don't know much about it, but you can find some articles on Yann Olivier's website, and Jürgen Jost and Christian Leonard gave a minicourse on it at the IHS last spring. These might be some starting points for you.

These notes by John Lott (covering some joint work with Villani) do it for length spaces, which finite metric spaces never are, but if you join the points by edges whose lengths are the distances (so topologically, you have a complete graph), then all is well, and you can use the machinery (which may or may not do what you want).