The Talosian system consisted of eleven planets orbiting the binary stars Talos A and Talos B. One of the planets, Talos IV, was inhabited by a technologically advanced species, the Talosians. Talos III had also been visited by the late 23rd century.
Eighteen years later, the USS Enterprise discovered the fate of the Columbia after being lured to the system by a faked distress call. After escaping the Talosians, the crew recommended that the planet be placed under quarantine, a recommendation that Starfleet Command endorsed. General Order 7 was created to prohibit contact between the Talosians and Federation starships. (TOS: "The Cage")
The Talosian system was at one point to have been a double star system. (The Making of Star Trek, pp. 93 and 96) Before production on that episode, Gene Roddenberry conceded that he was somewhat "at a loss [...] on exactly what it should look like," and further commented, "If it is too bizarre or confusing to the audience, we'll probably have to go back to a single sun." (The Making of Star Trek, p. 96) The map for the system showed two stars of equal size at the center of the star system; however, when the shore leave party was on the surface, it's not possible to see the suns as the sky was overcast.
In the "The Explored Galaxy" star chart, Talos was located between the Tholian Assembly and Romulus. Both were identified in DS9: "Call to Arms" as being located in the Alpha Quadrant. According to Star Trek: Star Charts, the Talos system was located in or near Federation space, in the Alpha Quadrant. ("United Federation of Planets III")