Gotham City has a “strong mayor” system of government, and it's charter is almost unique in how much authority it gives to the mayor, in relation to other city officials and the Board of Aldermen. Ironically, this was created to fight corruption, but has ended up facilitating it.
The head of the city government is the mayor who appoints all subordinate officers. The mayor appoints any number of Mayor's Aides who are issued a special badge, ordering police and city officials to render them all courtesy and assistance. The mayor also runs the parallel criminal organization called The City Hall Machine which exploits the government mechanisms for graft.
The roads, bridges, buses, port and subway are overseen by the The Gotham Transportation Authority and its senior administrator, all appointed by the mayor and the most corrupt organization in the city.
The Department of Relief is run by the Relief Commissioner who hires social workers to provide aid to the poor. About half of the social workers are no-show parasite relatives of gangsters, making relief work that much harder for the rest.
The District Attorney for Gotham, unlike the DA's in the rest of the state is appointed by the mayor.
There is an elected Board of Aldermen who must approve new tax measures and changes to city ordinances.