Quick - Regular - Epic - Marathon
Vanilla: Monasteries increase a city's cultural output if the city is near to improved Incense and/or Wine resources. The Monastery cannot be built unless the city has such an improved resource nearby.
Gods & Kings: Religious building. Can only be built in cities following a religion with the Monasteries belief. Can only be purchased with Faith.
Monasteries are unique to Religious buildings, because they don't boost Happiness as the other Religious buildings. On the other hand, if there are sources of Wine or Incense nearby, each of them receives a +1 bonus Culture and Faith potential! 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.
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.