In my opinion, Colin de Verdiere's book "Spectre de Graphes" is the best place to start investigating the connection between the discrete Laplacian and the manifold Laplacian.
Recently, investigations in computer science (machine learningtheory) have lead to considerable progress.
Pick a "cloud" of points in a Riemann manifold. Consider the complete graph with vertices on these points. Next add weights to the edges that correlate with the geodesic distance between the corresponding points on the manifold. Then form a certain weighted Laplacian associated to this weighted graph. This operator converges with probability one to the the manifold Laplacian as the cloud is bigger and bigger and is chosen randomly with respect to the metric volume measure on the manifold.
I know I skipped many details, but you can get precise statements at in this nice paper by M. Belkin and P. Niyogi:
M. Belkin's webpage has additional info.