A novel pure and N-doped waved-graphene electro-catalyst with various wavelengths for oxygen reduction reaction (ORR) is explored in acid medium based on the first-principles calculations. The associative and dissociative mechanism are both explored from the O adsorption to the HO formation step by step, from which the dissociative mechanism is likely the only way with an effective 4e process. Furthermore, a very favorable and effective mechanism is proposed for the ORR process, which may provide a guideline on the design of new non-metal electro-catalysts.