We consider regularized distributed optimization problems over networks. The problem arises from many existing domains, such as coordinated control, sensor fusion and distributed learning. We propose a new framework based on Bregman method and operator splitting, which allows us to come up with a general distributed algorithm, termed Distributed Forward-Backward Bregman Splitting (D-FBBS). The proposed algorithm, though derived from a different perspective, is shown to have close connections with some existing algorithms. In addition, we show that the proposed algorithm is able to seek a saddle point which solves the above primal problem as well as its dual simultaneously. We also establish a non-ergodic convergence rate of o(1/k) in terms of fixed point residual. A simple example of sensor fusion problem is provided to illustrate the effectiveness of the algorithm.