(covers information from several alternate timelines)
"Nobody but us chickens, doctor."
Delta Vega was located in the Delta Vega sector, a few light days away from the galactic barrier at impulse speed for a Constitution-class heavy cruiser. (TOS: "Where No Man Has Gone Before"; TNG: "Aquiel", okudagram)
In 2293, the location of Delta Vega in the Milky Way Galaxy was labeled in the star chart "The Explored Galaxy", which was on display in Captain James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In 2366, in an alternate timeline, during the Federation-Klingon War, the location of Delta Vega was labeled on a tactical situation monitor in the ready room aboard the USS Enterprise-D. (TNG: "Yesterday's Enterprise", okudagram)
Prior to stardate 1312, the Galactic Mining Company, working with the Federation, established the Delta-Vega Station on this planet. An automated lithium cracking station was established in a valley. This station was, at most, visited by ore ships every twenty years. Another valley was, from the station's perspective, to the left of some mountains with pointed peaks, beyond which flatlands were situated. In the same general direction was a small clearing, amid a rock-strewn area. Delta Vega's terrain was generally very rocky and the planet's weather was known to include winds, such as a dusty breeze during one day.
After the USS Enterprise suffered warp drive damage when crossing the galactic barrier in 2265, Delta Vega was visited by the crippled starship so that helmsman Lee Kelso and engineering crewmen could make repairs to the vessel. By the time of their arrival, the danger posed by the mutation of Gary Mitchell, via the barrier, into a continually strengthening psionic being was undeniable, and Captain Kirk planned to maroon his old friend on the planet. Mitchell objected, escaped, and fled into the hills of Delta Vega, taking Elizabeth Dehner with him.
Although Dehner thereafter opined that surviving on the planet would take "almost a miracle," the landscape was then miraculously seeded with vegetation, a result of Mitchell using some of his powers. For example, the small clearing became a garden ripe with plants (plentiful in flowers) around a little pool of water and a spring, due to Mitchell's intervention. Moments later, he also made a Kaferian apple tree appear in the same area.
The presence of both Kirk, who had meanwhile been tracking the pair, and Mitchell caused considerable disturbance in the area and a battle between them culminated in a load of rocks burying Mitchell in a grave he had conjured up for Kirk. After Dehner died (as a result of having fought against Mitchell herself), the remains of both Mitchell and Dehner were left behind on Delta Vega and – in an entry of the captain's log of the Enterprise, made while the ship left the planet – they were noted for having given their lives in the line of duty. (TOS: "Where No Man Has Gone Before")
Behind the scenes
In the script of "Where No Man Has Gone Before" (Star Trek's second pilot episode), this planet was commonly written with a hyphen between the words "Delta" and "Vega" (i.e. being notated as "Delta-Vega"), which corresponds with the sign at the cracking station indicating "Delta-Vega Station". The episode's first view of the planet's surface, showing its lithium cracking station, was described thus; "At the building, a suggestion of alien planet strangeness in the coloration of the soil, the weird vegetation in the planting beds next to the [...] building. Beyond that, the eerie surface of Delta-Vega." The script also specified that a "distant stretch of planet Delta-Vega" outside the station's main window was to be created on stage. The script characterized the aforementioned rock-strewn area, which Elizabeth Dehner and Gary Mitchell walk through prior to the latter creating the garden site, as "a weird, pitched and jumbled naked rock terrain, stretching off into the distance" and stated that it had "odd-shaped rock formations." The garden area was referred to as "a pool of water surrounded by thick, thick, fruit-bearing vegetation."
The interiors of the Delta Vega cracking station were built on Desilu Culver Stage 17, while Stage 16 housed the surface exteriors. Scenes on Delta Vega were filmed between Friday 23 July 1965 and Wednesday 28 July 1965, with extra pick-up shots filmed on Thursday 29 July 1965. (These Are the Voyages: TOS Season One)
The set that was used to depict the surface of Delta Vega had previously been used for planet Talos IV in Star Trek's first pilot, "The Cage", including the same backdrop and even some of the same rocks. (Starfleet Access for "Where No Man Has Gone Before", TOS Season 1 Blu-ray special features; "Where No Man Has Gone Before" text commentary, TOS Season 1 DVD special features) So that the pond could be set into the ground, the garden area was constructed on a slight platform. ("Where No Man Has Gone Before" text commentary, TOS Season 1 DVD special features)
As used for Delta Vega, the set was prepared for filming by 28 July 1965, the last day on which principal photography for "Where No Man Has Gone Before" was shot and a day on which the climactic battle scene between Kirk and Mitchell was filmed. However, there was a problem with the planet set; each take caused loose sand from the set to scatter across the camera dolly tracks, causing even more delays than there already had been. (Inside Star Trek: The Real Story, p. 85) Director James Goldstone offered, "The set was made largely of styrofoam. When the actors moved around the styrofoam it chipped and flaked, and it would get all over the dolly tracks. The sound man would say, 'You can't use that take,' because you could hear, 'Crunch, crunch, crunch.'" (The Star Trek Interview Book, pp. 110-111) The stage was cleaned up by Robert Justman and Herbert F. Solow after each take and, at one point, Desilu president and owner Lucille Ball also helped with the sweeping of the sand. (Inside Star Trek: The Real Story, p. 85)
The dossier for the official Star Trek movie website stated that, before the destruction of Romulus in the late 24th century, Nero and his mining operation discovered lithium deposits on a "Delta Vega". Since it was unlikely they would have been allowed to mine in the Vulcan system, it was possible that Nero was mining this Delta Vega, as it may have been abandoned by the Federation by that time.
For the pre-remastered version of "Where No Man Has Gone Before", Delta Vega was the first use of the brown/red model of planet. This model was used later for M-113 in ("The Man Trap"), Tantalus V ("Dagger of the Mind"), Planet Q ("The Conscience of the King"), Benecia ("The Conscience of the King"), Gothos ("The Squire of Gothos"), Lazarus' planet ("The Alternative Factor"), Beta III ("The Return of the Archons"), Janus VI ("The Devil in the Dark"), Organia ("Errand of Mercy"), Mudd ("I, Mudd"), Argus X ("Obsession"), and Ardana ("The Cloud Minders"). For the remastered version of "Where No Man Has Gone Before", a new planet model was created by CGI artist Max Gabl.
According to Star Trek Maps (p. 25), Delta Vega was the only planet orbiting a white dwarf star at the northern fringe of the galaxy. It was a class F planet with primitive life only and a primordial atmosphere.
According to Star Trek: Star Charts (pp. 10, 24) and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Delta Vega was a G-class planet in the Alpha Quadrant. The primary, Delta Vega, was an A-class star. The star and its system were located "above" the Federation.