A launch bay is a location from which vehicles, be they spaceborne or otherwise, are dispatched. On a starship, a launch bay is the area from which shuttlecraft and other small craft may be launched, and where they are docked when not in use. The term was in common use in the 22nd century but by the 23rd century, the term shuttlebay was more common.
Sometime before late 1996, Henry Starling constructed a launch bay within the uppermost floors of Chronowerx Industries' headquarters. This launch bay housed the Federation timeship Aeon, which Starling had appropriated in the year 1967. (VOY: "Future's End", "Future's End, Part II")
In the NX-class of starship, the launch bay opened from the ventral hull and was capable of holding two shuttlepods. These shuttlepods both entered and exited through two sliding doors, which were operated from a control station on the upper gantry level; the launch bay was incapable of maintaining atmospheric pressure during launch, requiring repressurization after every launch. The launch bay was equipped with a telescoping arm to guide craft through the bay doors. The ship's grappler was also housed on the ceiling of this bay. (ENT: "Cold Front", "Divergence")
Due to their compact size, the Defiant-class also utilized a launch bay, instead of the larger shuttlebays used on other Starfleet ships of the late 24th century. The USS Defiant's launch bay was used by Type 18 shuttlepods and Type-10 shuttlepods, such as the Chaffee. (DS9: "The Sound of Her Voice")
Dominion Headquarters on Cardassia Prime had at least three launch bays. A Cardassian patrol ship was sitting on launch bay 3-A in 2375. Damar told Worf and Ezri Dax of the launch bay's location, so they could escape execution by the Dominion. (DS9: "Strange Bedfellows")
The set of the Defiant launch bay was built in its entirety. Foundation Imaging employee Koji Kuramura additionally built a CGI version of the Defiant launch bay, for DS9: "The Sound of Her Voice". Even though Foundation did not work on many DS9 episodes, production illustrator Doug Drexler provided Kuramura with detailed drawings of the whole set, on which the latter production staffer based the CGI representation. Recalled Kuramura, "The only thing was, I could not quite figure out where to put the hanger doors. I had to do it so it didn't change the exterior of the Defiant model. Other than that it was a fun model to build." 
When Gary Hutzel asked Doug Drexler to design a large shuttlepod which was to eventually become the Chaffee type shuttlepod for the Defiant, Drexler was perplexed by where the craft could possibly launch from, having newly completed the diagrams of the Defiant for the Star Trek: Deep Space Nine Technical Manual. He recounted, "Gary! I blurted like a little kid who didn't wanna. Where? Where is it supposed to launch from! It's impossible! There is no place! And to prove it I unfurled the newly minted diagrams on top of the lightbox. See? I've worked out every nook! There is absolutely no place… no place… um… hey… it could go right here! I point to the circular ring on the bottom of the ship. Right here! Some modest adjusting and… it… could… work! There was another flash of light and a roiling of thunder and he was gone! His disembodied voice reverberated over me… Today…today… today…" Yet, Drexler was able to produce a concept sketch, rendered in Adobe, that was sent to Kuramura.  The renewed shuttlebay was justified by Drexler in the technical manual (p. 138) by stating, "The shuttlecraft bay has been built into Decks 3 and 4, immediately below the "Defiant" bridge, in a reserved volume originally designed for future computer and weapons systems."
The NX-class launch bay was designed by John Eaves, a task that was assigned to him by Production Designer Herman Zimmerman and overseen by Executive Producers Rick Berman and Brannon Braga. "There was no trouble getting that one through," Eaves said about the design. "The only changes we had to make were to do with the scale. It sized down considerably every time we drew it, but the basic elements remained the same. We had some elements from the engine room – we really liked the walls and the pipes. It was just the framework which really sold the whole set. They liked the idea of being able to walk around and look into the shuttles from the second floor, as opposed to being next to them on the floor." (Star Trek: The Magazine Volume 2, Issue 7, p. 52)
Another facet of the launch bay design that evolved was the idea of the horizontal doors built into the floor of the set rather than the doors being vertical, at the aft of the ship. Eaves explained, "Mike Okuda pushed [that] idea [....] He did a little computer sketch, and Mr. Berman really liked it. I did renderings on different doors. In the beginning it was going to be like a clamshell where it split in the middle and either of the shuttles could drop out. Finally, Doug [Drexler] redesigned the floor so now independent doors open under each shuttle." (Star Trek: The Magazine Volume 2, Issue 7, p. 55)
Though the NX-class was built as a full-scale set for Enterprise, the launching sequences of the shuttlepods had to be executed in CGI. The CG version of the launch bay aboard the NX-01 was built by Koji Kuramura, who remembered, "The building of the CG Set of Enterprise's Launch bay was never in the budget. I just thought it should be built, because I knew it would get used. Dougy helped me out by getting me a bunch of photographs and a tour of the set, so that I could build it for the Series premiere episode [namely, "Broken Bow"] [....] I built everything. From Ceiling to Floor, almost every nut and bolt. Went a little crazy."