The purpose of this paper is to establish an equivalence between mixed and nonconforming finite element methods for second order elliptic problems on both triangular and rectangular finite elements, and to provide an analysis of multigrid methods for both methods based on the equivalence. We first show that the linear system arising from the mixed method can be algebraically condensed to a symmetric and positive definite system for Lagrange multipliers using features of mixed finite element spaces and that the system for the Lagrange multipliers is identical to the system arising from the nonconforming method. Then we prove that optimal order multigrid algorithms can be developed for both methods. Two types of multigrid methods are considered in this paper. The first one makes use of the coarse-grid correction on nonconforming finite element spaces, while the second one has the coarse-grid correction step established on conforming finite element spaces. Finally, numerical examples are given to illustrate the present theory.