The mayor is the deity figure in charge of the city being governed.
By the Sim constitution, the SimCity mayor has absolute power over the entire city. The mayor's term in office is indefinite, as long as there is no loss of confidence (mainly negative city budget). The mayor has the power to control all government buildings, and all aspects of city funding (depending on the game). The mayor also has the power to demolish any building at will, be it government or private.
Depending on the game, the mayor may also have control over the city terrain, transforming and terraforming it at will. These powers may make the mayor godlike to some extent, depending on the game, the mayor can know personal information of its citizens like happiness, wealth level, and even criminal intent (if any). The mayor also has the ability to summon "natural" disasters, and can even have some control over it during course of the calamity. However, the power of the mayor has certain limits, as the mayor is powerless to stop a disaster once it has manifested, and the mayor's control does not extend beyond the city borders.