(covers information from several alternate timelines)
Babel was the code name for an inhabited neutral planet in the Alpha Quadrant. This planet was used by United Earth, and later the United Federation of Planets, as a venue for conferences, which were mediated in council chambers. (TOS: "Journey to Babel"; TAS: "The Counter-Clock Incident"; ENT: "Babel One")
In 2293, the location of Babel in the Milky Way Galaxy was labeled in the star chart The Explored Galaxy, which was on display in James T. Kirk's quarters aboard the USS Enterprise-A. (Star Trek VI: The Undiscovered Country, okudagram)
In 2366, in an alternate timeline, during the Federation-Klingon War, the location of Babel was labeled on a tactical situation monitor in the ready room aboard the USS Enterprise-D. (TNG: "Yesterday's Enterprise", okudagram)
In 2399, the location of Babel was labeled in a Federation star chart that was in Fleet Admiral Kirsten Clancy's office at Starfleet Headquarters. This planet was in or near to Federation space. (PIC: "Maps and Legends")
The planet was planned to host negotiations to settle trade disputes between the Andorians and the Tellarites and mediated by Starfleet on behalf of United Earth in 2154. However, the meeting was postponed due to tensions between the two powers caused by a raiding Romulan scout vessel in the area. (ENT: "Babel One", "United")
In 2268, on stardate 3850.3, the Babel Conference considered Coridan's admission to the Federation. Two things came of this conference: the Babel Resolutions and Coridan's admission to the Federation. (TOS: "Journey to Babel"; TAS: "The Pirates of Orion"; TNG: "Sarek")
In 2270, the USS Enterprise transported Commodore Robert April and his wife Sarah to Babel, where ambassadors from all Federation worlds had converged to honor April as he had reached his mandatory retirement age. (TAS: "The Counter-Clock Incident")
According to Star Trek: Star Charts (pp. 45,66, "United Federation of Planets I") and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Babel was located in the Wolf 424 system. The system's primary was a M-class star. In the late 24th century, it was an independent system.
According to Star Trek Maps (p. 24 and chart D), Babel was located in the Wolf 424 system. The Babel planetoid had three small moons called Zion, Ba, and Ruim. Babel had a diameter of 4,200 kilometers and orbited a red dwarf star. The Federation sourcebook identifies Babel as the twelfth planet in the Wolf 424 system.