Not yet mentioned is the interesting definition of Ricci curvature by
Yann Ollivier, a definition especially suited to discrete spaces, such as graphs.
His definition "can be used to define a notion of 'curvature at a given scale' for metric
spaces." For example, he shows how the discrete cube $\{ 0,1 \}^n$
behaves like $\mathbb{S}^n$ in having constant positive curvature,
and possessing an analog of the Lévy "concentration of measure"
(the mass of $\mathbb{S}^n$ is concentrated about its equator).
His definition is used in the recent (April, 2011) paper by Jürgen Jost and Shiping Liu: "Ollivier's Ricci curvature, local clustering and curvature dimension inequalities on graphs."
Here are two primary sources:
Y. Ollivier, Ricci Curvature of Markov Chains on Metric Spaces, J. Funct. Anal. 256 (2009), No. 3, 810-864.
Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains, in Probabilistic approach to geometry, 343-381, Adv. Stud. Pure Math., 57, Math. Soc. Japan, Tokyo, 2010.
Update (7Feb13). Noticed a this recent posting to the arXiv:
Warner A. Miller, Jonathan R. McDonald, Paul M. Alsing, David Gu, Shing-Tung Yau, "Simplicial Ricci Flow," arXiv:1302.0804
[math.DG].
