In this paper, we develop and analyze an adaptive finite volume algorithm for second order elliptic boundary value problems. First, we derive a residual type a posteriori error estimator, and then establish upper bounds and lower bounds in comparison with the exact error. Using certain relationship of local stiffness matrices between finite element and finite volume for the Poisson equation, we establish the discrete local lower bound of the error between solutions on two successive refinements. After proving several additional technique results including the quasi-orthogonality for two different finite volume solutions, we finally obtain the error reduction and convergence of the adaptive finite volume method.