The Talosian system was a planetary star system. The Talosian system was located within the Talos star group, which was a region of space in the Alpha Quadrant. This system consisted of binary stars and had eleven planets. Long range scans of the system showed that it was similar to the Sol system.
The binary stars were Talos A and Talos B. One of the planets, Talos IV, was inhabited by a technologically advanced species, the Talosians. Another planet, Talos III, was suitable for colonization by humanoid life-forms, and a Federation colony was founded on this world in 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")
Behind the Scenes
The Talosian system was at one point to have been a double star system. (The Making of Star Trek, pp. 93 & 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.
"The Explored Galaxy"
"The Explored Galaxy" was a map of charted space in the Alpha Quadrant. Talos 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 this 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".
The Talos system was located in or near Federation space, in the Alpha Quadrant. ("United Federation of Planets III")