The current Mayor of the City is Ed Lee. He was elected using a modified Instant Runoff Voting system.

Mayor Ed Lee was chosen by Gavin Newsom to complete his term.  Lee then won to become  43rd mayor of San Francisco. He was elected using a modified Instant Runoff Voting system.

Famous prior mayors include Willie Brown, who served for many years as Speaker of the California Assembly prior to being mayor, and Dianne Feinstein, current U.S. Senator.

Wikipedia has a list of all the mayors if someone wants to get it.