Green Bank Telescope 21 cm observations have revealed a faint, yet extensive H I cloud population surrounding the Andromeda galaxy (M31). The newfound objects are likely analogs to the high-velocity H I clouds seen around the Milky Way. At least 20 discrete features are detected within 50 kpc of the M31 disk, with radial velocities that are comparable to those of outer disk rotation. In addition, a filamentary "halo" component of at least 30 kpc extent is concentrated at the M31 systemic velocity. Some of the discrete features are organized into elongated systems with velocity continuity, suggestive of tidal streams. The discrete population can be characterized by a steep power-law distribution of number versus H I mass in the range between 105 and 107 M☉. The velocity line width of discrete clouds is correlated with the cloud H I mass such that if the clouds are gravitationally bound this implies a dark matter to H I mass ratio of ~100 : 1. Possible origins for the discrete and halo M31 features include a Local Group "cooling flow," tidal debris from recent mergers or interactions, and the gaseous counterparts of low-mass dark matter halos.