A monastery was a community where members of a particular religious tradition live and work apart from the rest of society, while dedicating themselves to the ideals of their religious tradition. A number of races throughout the galaxy have monasteries - including Humans, Vulcans, Bajorans, and Klingons.
A notable Human monastery is the one located in Ngari, Tibet. Jonathan Archer once visited the monastery, which was the oldest such institution that he ever visited. (ENT episode: "The Andorian Incident")
The Vulcan monastery at P'Jem was over 2,000 years old by the 22nd century. At some point, the Vulcan High Command hid a listening post to use against the Andorians - when this was discovered the Andorians responded by destroying P'Jem. (ENT episodes: "The Andorian Incident", "Shadows of P'Jem")
Shortly before Kahless died on Qo'noS, he said he would someday return. Kahless picked the location at random when asked where he would reappear - pointing at a star in the sky. (TNG novel: Kahless) The Klingons established a monastery on a planet orbiting the star - which was called Boreth - in which Klingons seeking spirtual guidance would travel, hoping to have a vision of Kahless. Seeing the decline of the Empire, the monks at Boreth created a clone of the original Kahless. After Kahless, son of Kahless was revealed to be a clone, Chancellor Gowron backed Worf's plan to install this clone as the ceremonial Emperor. (TNG episode: "Rightful Heir")
Bajorans also established a number of monasteries in which vedeks and other members of the Bajoran clergy lived. The former Kai Opaka Sulan stayed at a monastery in the city of Ashalla after she returned from the Gamma Quadrant, and helped guide Elias Vaughn through an orb experience when he came to stay at the monastery while on medical leave. (DS9 novel: Unity)