(covers information from several alternate timelines)
Alpha Centauri was an inhabited planetary system in the Alpha Quadrant. Located approximately four light years from Sol, this was a trinary system of three stars – Alpha Centauri A, Alpha Centauri B, and Proxima Centauri. (TNG: "The Naked Now" production art)
In 2293, the location of Alpha Centauri in the Milky Way Galaxy was labeled on 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 2364, the location of Alpha Centauri was labeled on 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)
In 2399, the location of Alpha Centauri was labeled on a star chart that was in Fleet Admiral Kirsten Clancy's office at Starfleet Headquarters. This system was located in or near Federation space. (PIC: "Maps and Legends")
When Kirk was captured in 1969 on Earth, and questioned on who he was by Lieutenant Colonel Fellini, a member of the US Air Force Air Police at the US 498th Airbase Group, Kirk explained that "the truth is, I'm a little green man from Alpha Centauri, a beautiful place. You ought to see it." (TOS: "Tomorrow is Yesterday")
Between 2078 and 2119, Humans founded an outpost on the planet Alpha Centauri. Among its residents in this formative period was Zefram Cochrane, who moved to the system from Earth after 2069. He left the system circa 2119 for an unknown destination. Prior to 2124, Alpha Centauri City was founded. (TOS: "Metamorphosis"; TNG: "The Neutral Zone", display graphic; ENT: "Future Tense") Alpha Centauri was among the earliest systems to be colonized by Humans in the early space explorations. The other known systems included Terra Nova's system and Vega's system. (ENT: "Fortunate Son", "Twilight")
In an alternate timeline, created by Jonathan Archer being infected by interspatial parasites, the Xindi destroyed Earth in 2154 and then all other Human settlements, including Mars, Alpha Centauri, and Vega colony. (ENT: "Twilight")
In 2268, Kirk argued in favor of having Sargon and his companions take over the bodies of the crew, and emphasized the benefits that it might possibly have on mankind, he explained, "they used to say if man could fly, he'd have wings. But he did fly. He discovered he had to. Do you wish that the first Apollo mission hadn't reached the moon or that we hadn't gone on to Mars or the nearest star?" (TOS: "Return to Tomorrow")
In the mid-24th century, Dr. Leah Brahms resided in this system. She was a student at the University of Alpha Centauri and graduated from this university with a Bachelor of Science in Subspace Physics, cum laude. (TNG: "Booby Trap", okudagram)
In 2365, after Doctor Katherine Pulaski was abducted by James Moriarty, Geordi La Forge believed she planned to "lead Data on a wild goose chase and then recount the story to everyone between here and Alpha Centauri." (TNG: "Elementary, Dear Data")
According to The Worlds of the Federation, the seventh planet in the system was the homeworld of a group of Humans, called Centaurians. Evidence was found that they were in fact originally Greeks transplanted to the planet in the 3rd century BC.
The association of Alpha Centauri as the fifth founding member was also referenced in the "It's Federation Day!" newspaper article, created as background material for Star Trek Generations. According to that article, Alpha Centauri had recently gained its independence. Titus Oleet, who signed the founding document of the Federation, was the system's ambassador.
According to Star Trek: Star Charts (pp. 19, 45, & 56; "United Federation of Planets IV") and Stellar Cartography: The Starfleet Reference Library' ("Federation Historical Highlights, 2161-2385"), the Alpha Centauri system, consisting of Alpha Centauri A (a G-class star) and Alpha Centauri B (a K-class star), was located in the Alpha Centauri sector (Sector 002), a region of space in the Beta Quadrant. In 2120, Zefram Cochrane departed from the Alpha Centauri colony on his final voyage. In the 22nd century, this colony was a destination on the Earth trade routes and traded with Earth and Vulcan. Alpha Centauri became a Federation founding member in 2161.
The chart naming stars within twenty light years of Sol was originally 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. Alpha Centauri was the terminus for two major commercial routes, Epsilon Eridani-Alpha Centauri and Sol-Alpha Centauri.
Alpha Centauri was misspelled as "Alpha Centuri" in the "The Explored Galaxy" star chart graphic seen in Star Trek: The Undiscovered Country.
In the real world, in 2012, a team of astronomers announced the discovery of Alpha Centauri Bb, an Earth-size terrestrial planet. In the years that followed, the discovery remained stubbornly unconfirmed. Until in 2015, another team of astronomers determined that this elusive ghost planet was all along nothing more than a measurement artefact. Attempts to locate other planets in the Alpha Centauri system have been, to date, unsuccessful.
William Shatner's Mirror Universe Trilogy (Spectre, Dark Victory, and Preserver) suggested that Zefram Cochrane lived on Alpha Centauri IV; likewise, his mirror counterpart lived on the mirror Alpha Centauri IV. Spock and his counterpart compared and contrasted the histories of the two universes and realized that the differences in the naming a lake on the planet (Lake Sloane in the primary universe, Lake Riker in the mirror universe) suggested that first contact may have been the point of divergence between the universes and that the divergence may have been caused by the presence of the Borg and the USS Enterprise-E, a possibility disproved by "In a Mirror, Darkly".
In the FASA role playing game the Alpha Centaurians were the first extraterrestrial civilization Humans encounter. The historic meeting occurred in 2042 when the Earth's United Nations space craft, the UNSS Icarus, utilizing a Bussard collector, arrived in the Alpha Centauri system. In 2048, Zefram Cochrane, an Alpha Centarian, sent his warp drive discovery to Earth. This contrasts with Star Trek canon material, which establishes that Cochrane was an Earth-born Human.
Like in the FASA game, the Last Unicorn role playing game, and the Decipher RPG, Alpha Centauri was already inhabited by a humanoid species when Terran explorers first ventured there in 2066. The Terrans traded their warp drive technology for Centaurian terraforming technology, to repair Earth's shattered ecosystem in the aftermath of World War III.