An NX-class captain's mess
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.
Dinner at the captain's mess in 2151
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 NX-01, Captain Jonathan Archer often enjoyed a meal with two or three of his senior officers, T'Pol, Trip Tucker, and Travis Mayweather, in the captain's mess. (ENT: "Broken Bow", "Terra Nova") The captain's mess was also used for official dinners such as with Vulcan Captain Vanik in 2151. (ENT: "Breaking the Ice") Tucker also dined with Captain Hernandez in the captain's mess aboard Columbia NX-02 while he was briefly assigned to that vessel. (ENT: "Affliction")
The set of the captain's mess was built on Paramount Stage 8 and 9. (Call sheets)
On Galaxy-class starships, the captain's mess was on Deck 2. It was equipped with a large buffet-style table. (TNG: "Sins of The Father", "Violations")
The Galaxy-class captain's mess was a redress of the observation lounge set. It had distinctive windows that may suggest a location at the very front of that deck, although they do not perfectly match the windows on the USS Enterprise-D miniature.
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")
The USS Bellerophon seen in DS9: "Inter Arma Enim Silent Leges" also had a kitchen in place of the captain's dining room, suggesting the modification may be a standard option aboard the Intrepid-class. The real-world reason for this was that the Bellerophon set was merely a re-use of the Voyager set.
Other contexts[]
Aboard Romulan Warbirds such as the IRW Khazara, the commander had a small dining room near the bridge, where she could dine with senior officers. (TNG: "Face Of The Enemy")
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")