The connection between random walks and electrical networks, e.g. as laid out by Doyle and Snell: http://arxiv.org/abs/math/0001057. This gives really slick proofs of many classical results about random walks (e.g. Polya's Theorem regarding recurrence vs. transience on lattices $\mathbb{Z}^n$, cover time, hitting time), and also generalizes them to other graphs.