Citizenship is separate from nationality; it means agreeing to abide by a particular city-state's laws. There are currently three places in which you man have citizenship: Ankh-Morpork, Bes Pelargic, and Djelibeybi.
To become a citizen in Ankh-Morpork, you must apply at the council building on Speedwell Street. After ten days the council can approve your citizenship request, if you are at least four days old and have a sponsor who is at least ten days old and a citizen.
Only citizens of Ankh-Morpork can own property there.
To become a citizen in Bes Pelargic, you must go to the Hall of Records on the Street of the Smiling Dog in the Pong Ping district. There, in the citizenship application room, you can "apply" for citizenship. You will then be given an exam consisting of questions about the Counterweight Continent. Questions are taken from both the MUD and the books, and are different each time; however there is no charge to take the exam and you can retry as many times as you like. You must get 16 questions correct to pass.
Only citizens of Bes Pelargic can own property there.
To become a citizen in Djelibeybi, you must apply at the palace at the north edge of the city.
Only citizens of Djelibeybi can own property there.
Dual citizenships are not allowed here, except in the case of ambassadors.