A religious order was a religious community.
A group of Vulcan monks formed an order who lived at the monastery at P'Jem. During the time of kolinahr, the order must remain in silent reflection throughout the lunar cycle. (ENT: "The Andorian Incident")
During Minister Jaro Essa attempted a coup d'etat of Bajor in 2370, he requested the public support of Vedek Winn Adami due to their similar stance to return to orthodox values. Winn doubted that her support would mean much, because she concluded that "support from the Vedek of an order so small as mine surely would not be significant." When Jaro offered to align with her order during her attempt to become the next Kai, explaining "think of the prestige of having the new leader of Bajor being a member of your order." (DS9: "The Circle")
A short time later, in 2370, following the crash of Kira Nerys and Jadzia Dax's Bajoran raider over Bajor, they were successfully recovered by searchers who were from Vedek Bareil Antos's order. Later, Bareil warned Kira that only members of religious orders were allowed to travel safely on the streets. To improvise, Kira and Dax wore Vedek attire in order to travel to the Chamber of Ministers unnoticed. (DS9: "The Siege")
Later that year, when Quark pitted Doctor Julian Bashir and Miles O'Brien to a racquetball match, with the proceedings going to the Bajoran monks, Quark attempted to tip the odds in O'Briens favor by ensuring Bashir would lose. Quark attempted to accomplish this by offering Bashir a "medicinal brew" under the guise that it would enhance his energy and vigor. The brew in question was said to be "an ancient secret of their order." (DS9: "Rivals")