Game Info
Vanilla
Bonus building. Requires an improved source of Wine or Incense.
- +2
Culture Culture in each tile with Wine or Incense worked by the city
Gods & Kings and Brave New World
Religious building. Can only be built in cities following a religion with the Monasteries belief. Can only be purchased with 
Faith Faith.
- +2 Culture Culture
- +2 Faith Faith
- +1 Culture Culture and Faith Faith in each tile with Wine or Incense worked by the city
Strategy
In vanilla Civilization V, Monasteries are an excellent source of 
Culture Culture for cities that have access to Wine and/or Incense.
Starting in Gods & Kings, Monasteries are unique to religious buildings, because they don't boost 
Happiness Happiness as the others do. On the other hand, if there are sources of Wine or Incense nearby, each of them receives a +1 bonus to Culture Culture and Faith Faith potential! When added to the Monastery's own bonus, this could add up to quite a lot of faith and culture from a single building! Of course, this bonus effect is wasted in cities that have neither of these resources.
The Monastery's bonus seems somewhat lesser than other religious buildings, but Monasteries are cheaper to purchase than other religious buildings, which frees up Faith Faith for other uses.
Historical Info
A monastery is a building or complex of buildings housing monks or nuns - people who have renounced worldly things (wealth, possessions, sex) - and dedicated their lives to prayer and meditation. Generally monasteries are single-sex organizations, or if men and women are present, they're separated by imposing walls. Historically monasteries have been places of higher learning, especially during the Middle Ages where members of religious orders were often the only people outside of the ruling elite who knew how to read and write.