In multicellular organisms and complex ecosystems, cells migrate in a social context. Whereas this is essential for the basic processes of life, the influence of neighboring cells on the individual remains poorly understood. Previous work on isolated cells has observed a stereotypical migratory behavior characterized by short-time directional persistence with long-time random movement. We discovered a much richer dynamic in the social context, with significant variations in directionality, displacement, and speed, which are all modulated by local cell density. We developed a mathematical model based on the experimentally identified "cellular traffic rules" and basic physics that revealed that these emergent behaviors are caused by the interplay of single-cell properties and intercellular interactions, the latter being dominated by a pseudopod formation bias mediated by secreted chemicals and pseudopod collapse following collisions. The model demonstrates how aspects of complex biology can be explained by simple rules of physics and constitutes a rapid test bed for future studies of collective migration of individual cells.