(covers information from several alternate timelines)
Alpha Centauri was an inhabited star system which also included the star Proxima Centauri. Other nearby stars included Sirius, Sol, and Tau Ceti. (TNG: "The Naked Now", production art; ENT: "Twilight")
In the early 22nd century, an Earth colony, the Alpha Centauri colony, was established on this planet. A major city of this colony was the Alpha Centauri City. (TNG: "The Neutral Zone", display graphic)
The location of the Alpha Centauri system in the Alpha Quadrant was depicted on two Federation star charts. A large star chart depicted the explored areas of the Milky Way Galaxy in 2293. (Star Trek VI: The Undiscovered Country, production art) It was also seen in a viewscreen chart displayed by the USS Enterprise-D library computer. (TNG: "The Naked Now", display graphic)
When James T. 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")
The Alpha Centauri colony was founded as an outpost on this planet in the early 22nd century. Over the decades, the colony grew in size and importance. Among its residents in this formative period was Zefram Cochrane, who moved to the system from Earth sometime following his historic test flight of the Phoenix in 2063. He left the system after 2119 for an unknown destination. He was presumed dead in space.
This planet was one of Earth's most important trade partners in the early days of Human interstellar exploration. It was one of the earliest colonies established, along with the Terra Nova colonies and the Vega colony. Eventually, the University of Alpha Centauri was founded on this planet. (TOS: "Metamorphosis", TNG: "Qpid", ENT: "Twilight")
In 2269, 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 another alternate timeline, the Xindi destroyed all Human outposts they could find, after destroying Earth, including the ones located on Mars, Alpha Centauri and Vega colony in 2154. (ENT: "Twilight")
|Alpha Centauri system|
|Alpha Centauri A • Alpha Centauri B • Proxima Centauri|
|Centauri VII • Proxima colony|
In the episode "Terra Nova", the first Earth-like planet identified by Human astronomers in the 2060s was located in a system that was 20 light years from the Sol system. This might suggest that the planet in Alpha Centauri might not be Earth-like.
According to Startrek.com, Alpha Centauri (Proxima) was located about 4.3 light years from Sol.  According to Star Fleet Technical Manual, the Concordium of Planets, located in the Alpha Centauri system, was a founding member of the United Federation of Planets. 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 3rd century BC.
The chart naming stars within 20 light years of Sol was originally drawn by Rick Sternbach for the Star Trek Spaceflight Chronology in the late 1970s. Found on page 77, this chart showed Earth commercial and exploration routes after the use of warp drive began. Alpha Centauri A was the destination on a major commercial route. This chart was, also, seen in the episodes "The Last Outpost" and "Conspiracy".
Alpha Centauri was misspelled as "Alpha Centuri" in the "The Explored Galaxy" star chart graphic seen in "Star Trek: The Undiscovered Country". "The Explored Galaxy" was a map of charted space in the Alpha Quadrant. This system was located between the Tholian Assembly and Romulus. Both locations were identified in DS9: "The Search, Part II", "Call to Arms" as being located in the Alpha Quadrant. This chart was also seen in several Star Trek: The Next Generation and Star Trek: Deep Space Nine episodes set in the 24th century, from the year 2364 to 2370. These were: TNG: "Conspiracy", "The Measure Of A Man", "The Emissary", "The Mind's Eye", "The Game" and DS9: "In the Hands of the Prophets", "Cardassians".
In 2012, astronomers announced the discovery of an extrasolar planet of comparable size to Earth orbiting close to Alpha Centauri B. The discovery strengthened the possibility that other Earth-like (M-class) planets may exist in the system.
Alpha Centauri: In the early 22nd century, an Earth colony was founded on a M-class world in this system. This world had the same name as the system. In 2120, Zefram Cochrane departed from Alpha Centauri on a final voyage. In the mid-22nd century, Alpha Centauri was a destination on the Earth trade routes. The colony traded with Earth and Vulcan. Alpha Centauri was a Federation founding member in 2161. In 2378, Alpha Centauri was a member planet. (pgs. 45, "United Federation of Planets I", "United Federation of Planets IV")
Alpha Centauri System: The system was located in or near Federation space. Alpha Centauri A was a G-class star with a magnitude of +5, which was the same brightness of Sol. Alpha Centauri B was a K-class star. This system was located in Sector 002 (Alpha Centauri sector). (Pgs. 19, 36)
Sector 002 (Alpha Centauri sector): This was the second sector to be explored by Earth ships. (Pg. 19)
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 universe 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 Centauran terraforming technology, to repair Earth's shattered ecosystem in the aftermath of World War III.