Proxima Centauri, also known as Alpha Centauri C, was a star in the Alpha Centauri trinary system. It was the closest star to Sol, located approximately 4 light years away. The star was located in the Alpha Quadrant. (TNG: "The Naked Now", "The Last Outpost", production art)
In 2364, the location of Proxima Centauri was labeled in a star chart of a stellar neighborhood with Sol at the center. This chart was stored in the USS Enterprise-D library computer. Later that year, the chart was scanned by Outpost 63. (TNG: "The Naked Now", "The Last Outpost", production art)
Apparently in an early draft script of TOS: "Wolf in the Fold", a Human colony established before 2156 was referenced as Alpha Proxima II, possibly intended as a planet in this system. The name was later replaced as Alpha Eridani II in the final episode. The Alpha Proxima II name, however, survived in such reference works as Star Trek Maps, Star Trek Concordance, the entry on StarTrek.com, and Star Trek Encyclopedia (3rd ed., p. 8).
A certificate naming the Proxima Centauri Interplanetary Bowling League was created for TNG: "The Schizoid Man", but did not end up used on screen. It was included in Star Trek: The Next Generation - The Continuing Mission (p. 76)
The chart naming stars within twenty light years of Sol was drawn by Rick Sternbach for the Star Trek Spaceflight Chronology (p. 77) in the late 1970s. This chart showed Earth commercial and exploration routes after the use of warp drive began. There were no commercial or exploration routes to Alpha Centauri C.
According to Star Trek: Star Charts (pp. 19, 45, & 56) and the Stellar Cartography: The Starfleet Reference Library' ("Federation Historical Highlights, 2161-2385"), the Proxima Centauri system, consisting of Alpha Centauri C (an M-class star), was located in the Alpha Centauri sector (Sector 002). In 2120, Zefram Cochrane departed from the Alpha Centauri system on his final voyage. In the 22nd century, this system was a destination on the Earth trade routes and traded with Earth and Vulcan.
According to Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Proxima (Alpha Centauri IV, Velestus) was an inhabited planet in this system. This planet had two moons, named Nykos and Tharon. The Proxima colony was founded on this planet in 2085. In 2161, this planet was a founding member of the UFP. In the late 24th century, the capital cities of this world were Cochrane and Tarola'n. The primary species of this planet was Human, with a population of 380 million.
In a real-world corollary to Star Trek's depictions, the discovery of an exoplanet orbiting Proxima Centauri, named Proxima Centauri b (lower case letters were used by Earth cosmologists in the early 21st century for planets orbiting a star) was announced in August 2016, by the European Southern Observatory from one of their South American observatories, by a team led by Guillem Anglada-Escudé. 
Further to this real-world analogy, later discoveries in the early 21st century suggested there could be two more planets in this star system. One a cold planet, in a further out orbit, designated Proxima Centauri c. The other, from a signal that was less well defined and so only seen as a potential candidate exoplanet, orbited inward of Proxima Centauri b. So, Proxima Centauri b, just like Alpha Proxima II in the Star Trek universe, began to look like it actually was the second planet orbiting out from its star.