The exact location of the colony is not known, although it may have been on a world orbiting the star Proxima and thus in very close proximity to Earth. In 2154, at a time when relations between United Earth and the Klingon Empire were tense, the Klingons were spotted near the colony and thought to be doing reconnaissance for a possible future strike against Earth.
Zefram Cochrane founded the Proxima colony, colloquially known as Alpha Centauri, departing Earth in 2101 with ships capable of Warp 1.8. Beginning around 2103, the world thrived and was terraformed, peacefully declaring independence approximately 2134, 20 years before the founding of the Coalition of Planets. Proxima's Ambassador Titus Oleet signed the United Federation of Planets Constitution on October 11, 2161. (ST reference: Federation: The First 150 Years)
Alternately, the colony was founded in 2085 on the planet Alpha Centauri IV (also known as Velestus). The planet had two moons, Nykos and Tharon. The colony became a founding member of the Federation in 2161. By 2385, the colony had two capitals, Cochrane and Tarola'n, and it had a population of 380 million individuals, primarily human. (ST reference: Stellar Cartography: The Starfleet Reference Library: Federation Historical Highlights, 2161-2385)