Evolutionary dynamics models have been mainly studied on homogeneous infinite populations. However, real populations are neither homogeneously mixed nor infinite. We investigate the stochastic evolutionary game dynamics on structured populations represented by graphs. We consider three simple graphs of finite number of vertices: the star, the circle and the complete graph. We present exact formulae for the fixation probability of a single mutant individual introduced into the graph and the speed of the evolutionary process, namely the mean time to absorption (either mutant fixation or extinction) and the mean time to mutant fixation. Through numerical examples we show the significant impact of the structure of the population, the population size and the payoff matrix on the above quantities.