The captain's dining room or captain's mess was an area aboard Starfleet vessels specifically designed for personal use by the captain in order to entertain guests and eat meals in private. It was generally equipped with food service by means of a kitchen, galley, or replicators. The mess sometimes included the services of a steward, who brought the meals and drinks to the captain's table.
The NX-class had a captain's dining room just off of the ship's mess hall. It was served by the ship's chef. On Enterprise, Captain Jonathan Archer often enjoyed a meal with his two senior officers, T'Pol and Charles Tucker, in the captain's mess. (ENT: "Broken Bow") Tucker would also dine with Captain Erika Hernandez in the captain's mess during his brief stint on Columbia. (ENT: "Affliction")
Some Intrepid-class ships had a captain's dining room directly behind the mess hall, cabin 125-A on Deck 2. During the USS Voyager's first months in the Delta Quadrant, Neelix converted the captain's dining room into a kitchen area, much to Captain Kathryn Janeway's initial dismay. (VOY: "Phage")
Aboard Annorax's Krenim weapon ship in an alternate timeline, there was also a small dining room where Annorax entertained Tom Paris and Chakotay with selections of cuisine from worlds that his temporal incursion had obliterated. (VOY: "Year of Hell, Part II")