We present an analog to classic potential theory on weighted graphs. With nodes partitioned into exterior, boundary and interior nodes and an appropriate decomposition of the Laplacian, we define discrete analogues to the trace operators, the single and double layer potential operators, and the boundary layer operators. As in the continuum, these operators can represent exterior or interior harmonic functions with different boundary conditions. The formalism we introduce includes a discrete Calderón calculus and brings some well known results from potential theory to weighted graphs, e.g. on the spectrum of the Neumann-Poincaré operator. We illustrate the formalism with a cloaking strategy on weighted graphs which allows to hide an anomaly from the perspective of electrical measurements made away from the anomaly.