The Alpha Leonis system was an inhabited star system. Regulus was the primary of the system. The planetary system consisted of least three planets. Notable forms of life in this system included the eel-birds of Regulus V and the Regulan bloodworms. (TOS: "The Cage", "The Menagerie, Part I"; TNG: "The Vengeance Factor"; Star Trek VI: The Undiscovered Country, production art; TOS: "Amok Time"; ENT: "Two Days and Two Nights") The system was located three hundred light years from the Bajoran system. (DS9: "Fascination")
In 2154, a Vulcan High Command attack fleet massed in this system in preparation for an abortive preemptive strike against the Andorians. Regulus was the closest accessible star known to be outside the maximum range of Andorian Imperial Guard listening posts. (ENT: "Kir'Shara")
In 2293, the location of Regulus in the Milky Way Galaxy was depicted on the Federation star chart The Explored Galaxy. This chart was first seen on aboard the starship USS Enterprise-A. In the 2360s, the chart was seen again aboard the USS Enterprise-D, the outpost Starbase 173, and at Starfleet Headquarters on Earth. (Star Trek VI: The Undiscovered Country production art; TNG: "Conspiracy", "The Measure Of A Man", "Emissary")
The association between the Alpha Leonis system mentioned in "The Vengeance Factor" and the Regulus star and planets is based on reference material and the real world association. According to Star Trek Maps (pg. 29), The Worlds of the Federation (pg. 130) and Startrek.com, Alpha Leonis was the same system as Regulus. In the real world Alpha Leonis is the Bayer classification name for Regulus. 
According to the Federation star chart The Explored Galaxy, Regulus, the primary of its system, was located between 347 and 348 degrees. A NASA map of the galaxy indicated that Regulus was located northeast of the galactic core. A star chart of the galaxy, first seen in TNG: "The Emissary", showed that the explored portion of the galaxy had changed its position to the south of the galactic core. Based on the latest information from Star Trek Into Darkness, and assuming that Regulus, if its location in the first chart was accurately depicted in its relation to Sol, this star was located in the Alpha Quadrant.
According to the Star Trek: Star Charts (pgs. 56,61,62), the Regulus (also known as Alpha Leonis) system was located in the Beta Quadrant. It was a trinary star system. Regulus A was a B-class star with a magnitude of +1, which was a hundred times brighter than Sol. Regulus B was a M-class star. Regulus C was a K-class star. In 2152, this system was in non-aligned space. By 2378, the system had been incorporated into Federation space, and was a destination on the major space lanes.