The Vice Mayor's role is to step into the office of Mayor should the Mayor leave the office for any reason. These reasons include the end of the Mayor's term before an election is complete (which happens if a second election round is needed), resignation, or the death of the Mayor.
The Vice Mayor is appointed by the Mayor, but once appointed cannot be removed by the Mayor. Requirements to be appointed:
Apart from waiting to be promoted to Mayor, a Vice Mayor has no actual power or work to do.
Historically Mayors began to appoint Vice Mayors since several cities were left without government after some of the first elected Mayors resigned. (See: Developer News 2006-11-25 Vice Mayor)