What is the criteria for holding the position of mayor in a city?
Each individual city sets the rules for qualifications to run for major.
You should check with your city hall to determine what is required.
The head of city government is usually an elected mayor. In the US the idea was proposed to take city administration out of politics, hire a professional manager to do the work, and leave the mayor in a ceremonial position, not unlike the king in a constitutional monarchy. Most cities still leave the job to the mayor.
There are two US Senators from California, Dianne Feinstein (D-CA) and Barbara Boxer (D-CA). There are also 40 state senators in California, who are part of the state legislature. Each incorporated city in California has a mayor, but there is no such position as "Mayor of California." The mayor of Los Angeles, the largest city in California, is Eric Garcetti (as of 2014).