Nonnegative matrix factorization has been widely applied in face recognition, text mining, as well as spectral analysis. This paper proposes an alternating proximal gradient method for solving this problem. With a uniformly positive lower bound assumption on the iterates, any limit point can be proved to satisfy the first-order optimality conditions. A Nesterov-type extrapolation technique is then applied to accelerate the algorithm. Though this technique is at first used for convex program, it turns out to work very well for the non-convex nonnegative matrix factorization problem. Extensive numerical experiments illustrate the efficiency of the alternating proximal gradient method and the accleration technique. Especially for real data tests, the accelerated method reveals high superiority to state-of-the-art algorithms in speed with comparable solution qualities.