(covers information from several alternate timelines)
In 2155, representatives from Tellar took part in talks on Earth to form a Coalition of Planets. In 2161, Tellar became a founding member of the United Federation of Planets. (ENT: "Demons", "Zero Hour")
In a commercial for Tribbles cereal, Tellar was shown on a computer screen of a food synthesizer in the mess hall of the USS Ravenous. It was listed along with other locations such as Earth, Vulcan, Izar, Cestus, Andoria, Rigel VI, and Rigel X. (ST: "The Trouble with Edward", credit cookie)
While conceiving TOS: "Journey to Babel", D.C. Fontana wrote, in a developmental document, "Tellar is a savagely cold planet." The extreme coldness of the planet was intended to account for the Tellarites having lots of fur on their bodies. While this is reminiscent of what the Andorian homeworld would turn out to be, the only canon depiction of Tellar Prime in ENT: "Babel One" shows at least a portion of the planet to be seemingly temperate, with at least one open body of water (which is actually a depiction of southern Lake Michigan) among what appears to be green vegetation.
According to Star Trek: Star Charts, (pp. 18, 19, 36, & 44) and Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), the Tellar (61 Cygni) system was located in the Tellar Sector (Sector 007), a region of space in the Alpha Quadrant. The system was a binary system with a pair K-class stars. In the 22nd century, this system was a destination on an Earth trade route.
According to Stellar Cartography: The Starfleet Reference Library ("Federation Historical Highlights, 2161-2385"), Tellar was a planet orbiting Tellar A (61 Cygni A). The planet had two moons, named Kera and Phinda. Its official name was United Planets of Tellar Prime. Alternative names for this planet were Tellar Prime, Miracht, Pangeos V. The planet's capital was Shirgat. Tellarites were the primary species. In the late 24th century, the planet had a population of 2.3 billion. This planet was warp-capable in 1959.
The Worlds of the Federation reference book (p. 20) states Tellar to be the fifth planet of the 61 Cygni system. It also gives Miracht as an alternative name for the planet and lists two moons called Kehra and Phinda.
The Star Trek Star Fleet Technical Manual (T0:00:01:19) lists the United Planets of 61 Cygni as one of the founding members of the Federation.