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The mayor is a person who is in charge of the administration of a city. Sometimes, mayors get elected by the people who live in that city, at other times, a central government body may determine who the mayor of a certain city is.
In a federal countries like Germany the mayor can also head of the government of a city state. The Mayor of Hamburg runs the government just like the Minister President of Bavaria runs the Bavarian government. If Washington DC in the USA ever becomes a state the Mayor of Washington DC would be the Governor.