A patrol is the term used to describe a starship or group of people assigned to monitor a specific area for a period of time. Vessels specifically designed for this type of task were designated patrol ships.
In 2266, the USS Enterprise picked up Lenore Karidian from Planet Q. As part of the Karidian Company of Players, Karidian offered to perform Hamlet for the crew of the Enterprise in exchange for passage to the planet Benecia. Captain James T. Kirk accepted her offer, noting the crew had been on patrol for quite some time and could use the break in monotony. (TOS: "The Conscience of the King")
On stardate 5027.3, Enterprise Chief Medical Officer Doctor Leonard McCoy noted in his log his concerns about Captain Kirk's increase in tension and emotional stress. The Doctor could find no reason for this increase other than the ship had been on patrol for too long without relief or diversion. (TOS: "The Enterprise Incident")
In 2269, Doctor Sevrin and his followers managed to gain control of the USS Enterprise and set it on a course for the planet Eden. As the course would take the ship directly through the Romulan Neutral Zone, Captain Kirk asked Commander Spock to scan for Romulan patrols, fearing the Romulans would see it as an act of war. Fortunately for the Enterprise, there were no patrols at that time. (TOS: "The Way to Eden")
In 2369, a fleet of fifteen starships, including the USS Enterprise-D, USS Crazy Horse, USS Agamemnon and USS Gorkon, were assigned to patrol a region of space near to Ohniaka III, following a Borg attack on the planet. (TNG: "Descent")
In 2373, Starfleet assigned the USS Enterprise-E to patrol the Romulan Neutral Zone, seemingly in case the Romulans decided to take advantage of another Borg incursion into Federation space. In truth, this due to their lack of confidence in Captain Jean-Luc Picard who had previously been assimilated by the Borg. While on patrol, the ship's first sensor sweep detected twenty particles of space dust per cubic meter, fifty-two ultraviolet radiation spikes and a class two comet. (Star Trek: First Contact)
The Federation Naval Patrol was a military organization dedicated to ensuring security and exploring the oceans of member worlds. As a result of his childhood fascination with stories about the ocean and ancient sailing ships, Tom Paris planned to join the Naval Patrol upon graduating from high school. Paris's father, however, who was an admiral in Starfleet, disallowed this, and pressured Tom to join Starfleet instead. (VOY: "Thirty Days")