The locomotion of most fish and birds is realized by flapping wings or fins transverse to the direction of travel. Here, we study experimentally the dynamics of a wing that is flapped up and down but is free to move in the horizontal direction. In this table-top prototype experiment, we show that flapping flight occurs abruptly at a critical flapping frequency as a symmetry-breaking bifurcation. We then investigate the separate effects of the flapping frequency, the flapping amplitude, the wing geometry and the influence from the solid boundaries nearby. Through dimensional analysis, we found that there are two dimensionless parameters well describe this intriguing problem that deals with this fluid-solid interaction. The first one is the dynamical aspect ratio that combines four length scales, including the wing geometry and the flapping amplitude. The second parameter, the Strouhal number, relates the vertical flapping speed and its resultant forward flight speed. Overall, we emphasize the robustness of the thrust-generating mechanisms determining the forward flight speed of a flapping wing, as observed in our experiments.