It varies by juristiction. For most places in the United States, there are really only three requirements: (1) you are a US Citizen, (2) you are a legal adult, and (3) you are a permanent resident of...
(1) you are a US Citizen, (2) you are a legal adult, and (3) you are a permanent resident of the entity you wish to become mayor of (that is, you are a resident of the city/town/etc. where you are...