A monk is a member of a religious order. Traditionally, this term is applied to a person who lives in a monastery.
Monks were known in such races as Vulcans, Bajorans, and Humans (during their past history, and imitated by Q during an appearance to the crew of the USS Enterprise-D). (TNG: "Hide and Q")
On Earth, Tibetan Buddhist teaching monks were known as lamas. Doctor Phlox spent two weeks in the mid-22nd century with the high lamas studying Buddhism and learning to sing chords. (ENT: "Cold Front")
In the Bajoran religion, Vedeks, Prylars and Ranjens are also called monks. (DS9: "Emissary", "In the Hands of the Prophets", "The Collaborator", "The Darkness and the Light", "The Reckoning")
In late 2369, Vedek Bareil Antos told Commander Benjamin Sisko that he knew a monk when he was five years old who could "squeeze the pagh right out of your ear with his thumb and forefinger". (DS9: "In the Hands of the Prophets")
In a holoprogram set in 16th century Florence, Leonardo da Vinci had to deliver a bronzetto to the monks at Santa Croce and asked Catarina to accompany him and pray there. (VOY: "Scorpion")