The Milky Way Galaxy was commonly divided into four equally sized, cubic quadrants, defined by one meridian passing through the galactic core and a second one perpendicular to the first, that also passes through the galactic core. These four quadrants are referred to as the Alpha Quadrant, Beta Quadrant, Gamma Quadrant and Delta Quadrant. The United Federation of Planets and its neighboring powers, including the Klingon and Romulan Empires, were located in the Alpha and Beta Quadrants, whereas the Borg Collective was located in the Delta Quadrant, and the Dominion in the Gamma Quadrant. (Star Trek VI: The Undiscovered Country; ENT: "Regeneration"; DS9: "Rules of Acquisition")
The division into four quadrants was used by United Earth and the Vulcans as early as the mid-22nd century. (ENT: "Shockwave, Part II", "Regeneration") In 2152, after less than a year into the mission of Enterprise NX-01, it was remarked by Vulcan Ambassador Soval, that he considered the ship to be a danger to the quadrant. By that time, Enterprise had involved itself in several armed conflicts and escalated conflict between the Vulcans and the Andorians. (ENT: "Shockwave, Part II") In 2153, a subspace message sent by the Borg was determined to have a destination deep inside the Delta Quadrant. (ENT: "Regeneration")
In 2267, when Lazarus and anti-Lazarus began changing places between each other, the physical laws of the universe were distorted for a brief moment. Captain Kirk noted in his log that the effects were felt throughout the entire quadrant of space. According to Commodore Barstow of Starfleet Command, the effect occurred in every quadrant of the Galaxy and far beyond. (TOS: "The Alternative Factor")
Smaller areas of space in the Galaxy were also referred to as quadrants during the 23rd and 24th centuries. These included: Quadrant 9, Quadrant 448, Quadrant 904, the Morgana Quadrant, and the Drema Quadrant. (TOS: "The Squire of Gothos", "The Deadly Years"; TNG: "When The Bough Breaks", "Heart of Glory", "The Child", "Where Silence Has Lease", "Pen Pals")
In 2267, following the loss of the Federation shuttlecraft Galileo in Murasaki 312, the Enterprise shuttlecraft Columbus conducted a search of quadrant 779X by 534M. After reporting negative results, the shuttle was ordered by Captain Kirk to proceed to the next quadrant. (TOS: "The Galileo Seven") Later that year, negotiations to build a treaty port on Vendikar for the quadrant where the planet was located were conducted by Federation Ambassador Robert Fox. (TOS: "A Taste of Armageddon") During the same year, unit XY-75847 reported a fleet of Klingon ships in their sector, which was located in the same quadrant as the planet Organia. (TOS: "Errand of Mercy") Also during the same year, the planet Argelius II was of strategic importance as a space port to the Federation. At the time, it was the only port in that quadrant. (TOS: "Wolf in the Fold")
By 2268, a particular quadrant of space, which included such notables as Deep Space Station K-7, Donatu V, and Sherman's Planet, had been under dispute between the Federation and Klingon Empire since initial contact. In that year, Nilz Baris served as the Undersecretary for Agriculture in said quadrant. (TOS: "The Trouble with Tribbles")
Also in 2268, during a transport mission to Babel, the Enterprise encountered an unidentified vessel that was shadowing the Enterprise's route. Following contact with Starfleet, the Enterprise received acknowledgment which confirmed that there were no other authorized Federation vessels in the quadrant. (TOS: "Journey to Babel") During the same year, the Enterprise was exploring an outer quadrant of the Galaxy when it received a distress call from the planet Scalos. (TOS: "Wink of an Eye")
In 2269, the Enterprise was ordered to respond to a botanical plague that was devastating a planet in the quadrant of the Galaxy where the ship was operating at the time. (TOS: "The Cloud Minders") In 2270, when the Enterprise encountered Kukulkan's starship, it was noted that Starfleet listed no other authorized vessels in that particular quadrant. (TAS: "How Sharper Than a Serpent's Tooth")
In 2284, when the Enterprise was ordered to Regula I by Starfleet Command, Admiral Kirk informed Starfleet Command that "all we had was a boatload of children," but the craft was nevertheless diverted because it was the only ship in the quadrant. While en route to Regula, the Enterprise encountered the USS Reliant, under the command of Khan Noonien Singh. Once the two ships were in visual range of each other, it was noted that the Reliant had entered the same section of the quadrant as the Enterprise was currently in. (Star Trek II: The Wrath of Khan)
In 2364, the planet Aldea was discovered in quadrant 1, mark 9-0, in the Epsilon Mynos system, by the USS Enterprise-D. (TNG: "When The Bough Breaks") Later that year, the Enterprise-D received a communique from Starfleet reporting a battle in Quadrant 9, coordinates 070-mark-3, in the Klingon Neutral Zone. (TNG: "Heart of Glory") During that year, the Enterprise-D picked up an unusual disturbance in a nearby quadrant, more specifically in Sector 63. The disturbance was caused by the destruction of the USS Horatio. (TNG: "Conspiracy")
As of 2365, the Morgana Quadrant was a section of the Galaxy that had not yet been visited by a manned Federation vessel. (TNG: "Where Silence Has Lease") Talos IV was located in the third quadrant of Vernal Galaxy. (TOS: "The Menagerie, Part I")
Background information Edit
The minor quadrants were clearly of variable sizes. In some cases, a quadrant was an area outside the ship in visual range, such as in Star Trek II: The Wrath of Khan and "When The Bough Breaks", or an area that could still be quickly explored with sublight shuttlecraft, such as those mentioned in "The Galileo Seven". In other cases, quadrants were composed of multiple sectors, such as the ones mentioned in "Errand of Mercy" and "Conspiracy". In "Heart of Glory", "Quadrant 9, coordinates 070-mark-3" was also shortened as "Quadrant 907-mark-3".
According to the Star Trek Encyclopedia (1st ed., p. 11), the reason for splitting the Federation between Alpha and Beta Quadrants was to rationalize Kirk's line in Star Trek II: The Wrath of Khan that the Enterprise was the only starship in the quadrant. Quadrants and sectors have been used inconsistently in various Star Trek episodes and films. During The Original Series, the term "quadrant" was used rather freely, as was the term "sector". At times, "quadrant" seems to refer to a fourth of the entire galaxy, while at others it seems to be a portion of a smaller region. It was not until the 1989 episode "The Price" of Star Trek: The Next Generation that the current system of Alpha, Beta, Delta, and Gamma quadrants was firmly established. Retroactively, Star Trek VI: The Undiscovered Country adhered to this system as well, as did Star Trek: Enterprise.
Furthermore, according to the Star Trek Encyclopedia, Earth marks the border between the Alpha Quadrant and the Beta Quadrant. The display graphic on a PADD in Star Trek: Insurrection would seem to confirm this.   Dialogue in Star Trek: Voyager and Star Trek: Deep Space Nine, however, firmly establish that Earth is located on the Alpha Quadrant side of the border.
It is unclear how the names for each of the four major quadrants were assigned, as they do not seem to follow any discernable pattern, instead of in clockwise or counterclockwise order. Viewing the galaxy from a top-down map with the "Alpha" quadrant in the lower left, and going around clockwise, the order is Alpha, Gamma, Delta, Beta.
In VOY: "Favorite Son", Taymon claimed he wanted to explore every "quarter" of the galaxy, suggesting his civilization similarly divided the galaxy in four, but termed the resultant segments differently.
In the third season edition of the Star Trek: The Next Generation Writers' Technical Manual (p. 16), the authors of this reference work wrote that a sector "was composed of four quadrants."