Synthesizing bilayer graphene (BLG), which has a band gap, is an important step in graphene application in microelectronics. Experimentally, it was broadly observed that hydrogen plays a crucial role in graphene chemical vapor deposition (CVD) growth on a copper surface. Here, by using ab initio calculations, we have revealed a crucial role of hydrogen in graphene CVD growth, terminating the graphene edges. Our study demonstrates the following. (i) At a low hydrogen pressure, the graphene edges are not passivated by H and thus tend to tightly attach to the catalyst surface. As a consequence, the diffusion of active C species into the area beneath the graphene top layer (GTL) is prohibited, and therefore, single-layer graphene growth is favored. (ii) At a high hydrogen pressure, the graphene edges tend to be terminated by H, and therefore, its detachment from the catalyst surface favors the diffusion of active C species into the area beneath the GTL to form the adlayer graphene below the GTL; as a result, the growth of BLG or few-layer graphene (FLG) is preferred. This insightful understanding reveals a crucial role of H in graphene CVD growth and paves a way for the controllable synthesis of BLG or FLG. Besides, this study also provides a reasonable explanation for the hydrogen pressure-dependent graphene CVD growth behaviors on a Cu surface.