A monk was a member of a religious order. Traditionally, this term was applied to a person who lived in a monastery.
Monks were known in such races as Vulcans, Bajorans, Humans, and Kadi and imitated by Q during an appearance to the crew of the USS Enterprise-D in 2364. (ENT: "The Andorian Incident"; TNG: "Hide and Q"; DS9: "In the Hands of the Prophets"; VOY: "Someone to Watch Over Me")
In the Bajoran religion, vedeks, prylars, and ranjens were also called monks. (DS9: "Emissary", "In the Hands of the Prophets", "The Collaborator", "The Darkness and the Light", "The Reckoning")
In 2365, Lieutenant Commander Data told a joke to Guinan which began as "a monk, a clone, and a Ferengi decide to go bowling together". (TNG: "The Outrageous Okona")
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")
Silent monks were mentioned in the Vulcan tale of enlightenment "Falor's Journey". (VOY: "Innocence")
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")