Someone who asks people for money in a public place is a panhandler. The term is fairly derogatory, but it's commonly used for those whose mainly support themselves this way.
It's best to avoid vaguely offensive terms like panhandler and beggar altogether — people asking for handouts have identities beyond begging for money or food. Still, it's good to know what panhandler means, and interesting to guess at its origin. Experts are split on this, believing it may stem from the shape of a pan, its handle extended like a begging arm; or that it arose from pans used as containers for people to toss change in.