In this paper, we investigate a Bogoyavlenskii-Kadomtsev-Petviashili equation, which can be used to describe the propagation of nonlinear waves in physics, biology and electrical networks. We find that the equation is Painleve integrable. With symbolic computation, Hirota bilinear forms, solitary waves and multi-front waves are derived. Elastic collisions between/among the two and three solitary waves are graphically discussed, where the waves maintain their shapes, amplitudes and velocities after the collision only with some phase shifts. Inelastic collisions among the multi-front waves are discussed, where the front waves coalesce into one larger front wave in their collision region.