We describe experiments that show collective behavior in clouds of optically trapped neutral atoms. This collective behavior is demonstrated in a variety of observed spatial distributions with abrupt bistable transitions between them. These distributions include stable rings of atoms around a small core and clumps of atoms rotating about the core. The size of the cloud grows rapidly as more atoms are loaded into it, implying a strong long-range repulsive force between the atoms. We show that a force arising from radiation trapping can explain much of this behavior.