- the palace (in Rome) which is the principal official residence of the Pope (i.e., the Head of the Roman Catholic Church} - as in ‘The Vatican is attached to the Basilica of St. Peter and contains administrative offices, a library, and a museum’ or ‘Every Sunday the Pope blesses the pilgrims in St. Peter’s Square from a balcony of the Vatican.’
- the Pope and/or other senior officials of the Roman Catholic Church - as in ‘The Vatican has recently declared the death penalty to be inadmissible in all circumstances’ or ‘The Vatican has softened its opposition to artificial forms of birth control.’
- the state whose territory forms an enclave within the city of Rome on the right (i.e., west) side of the river Tiber - as in ‘The Vatican has diplomatic relations with most countries in the world’ or ‘The population of the Vatican is about 1,000’. The official name of this state is the State of Vatican City (in Italian: Stato della Città del Vaticano; and in Latin: Status Civitatis Vaticanae); and it is probably better when speaking about the Vatican as a political or geographical entity to refer to it as Vatican City or the Vatican City - the use of the definite article is optional here.
Vatican is also used as a modifier - as in ‘The manuscripts of many classical authors are to be found in the Vatican Library’, ‘The Vatican position on a number of social issues has changed in recent years’ or ‘A Vatican spokesman has confirmed that the Pope will speak to the Bolivian president tomorrow.'
For more about the Vatican see Vatican – history.