A monastery was a religious retreat, usually isolated, and normally home to monastic groups devoted to the ideals of their order. Monasteries were prevalent throughout the races of the Alpha and Beta Quadrants, particularly among those species with more spiritual societies.
In June of 2151, Captain Jonathan Archer claimed that the oldest monastery that he ever visited was in Ngari, Tibet on Earth. It was significantly smaller then the Vulcan P'Jem monastery. (ENT: "The Andorian Incident")
While visiting Earth during the mid-22nd century, Doctor Phlox attempted to familiarize himself with many of the planet's religions including Buddhism, which he accomplished by spending two weeks at a Tibetan monastery where he learned to sing chords with the high lamas. (ENT: "Cold Front")
The Vulcan monastery at P'Jem on an unnamed planet near the Andorian system was already a thousand years old by the 22nd century and was still used by Vulcan monks prior to its destruction by the Andorians in 2151. (ENT: "The Andorian Incident", "Shadows of P'Jem")
The Klingon Followers of Kahless maintained monastery at Boreth. These individuals were considered the most devout followers of Kahless the Unforgettable. In the 23rd century no one ever left the monastery after being admitted, and no one who did not belong ever gained admission. At L'Rell's request, Ask Tyler left the son she and Voq had conceived at the monastery to be raised by the monks. (DIS: "Point of Light"). The monastery was central to the legend of Kahless the Unforgettable and his promised return. In 2369, Worf visited the monastery in an attempt to rediscover his Klingon heritage. It was there that Kahless apparently returned from Sto-vo-kor, although it was later revealed that he was a clone. (TNG: "Rightful Heir") Worf visited the planet again in 2371 following the destruction of the USS Enterprise-D. (DS9: "The Way of the Warrior")