A simple, unified theory of normal mode analysis of networks is presented. Unlike previous treatments, the present treatment does not rely on any particular method of formulation of equilibrium equations, nor on any detailed results of matrix theory. Rather, it is based on Tellegen's theorem, an elementary consequence of the Kirchhoff laws. Using the latter, together with a uniqueness theorem and a result concerning the number of modes in a network, all of the usual results of the normal mode analysis are obtained, namely, the decomposition of free and forced behavior into normal modes and the equations of motion of a set of normal coordinates and their relation to the network variables. The results are used to calculate the equations governing the behavior of coupled networks. While the detailed results are developed for the RLC network, the general nature of the approach allows extension of the results to a much wider class of networks and systems.