This paper studies the local bifurcation direction, stability properties and global structure for a nonlinear pseudodifferential equation, which describes the periodic travelling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. We first obtain the precise formula of the second derivative of bifurcation parameters at the bifurcation points. In particular, their signs can be strictly judged when constant vorticity vanishes. Furthermore, we present the stability analysis for the travelling water waves that have small vorticity and amplitude. We also show that the global bifurcation curves can't form a loop. Moreover, if the total head is bounded, the existence of waves of all amplitudes from zero up to that of Stokes' highest wave has been established.