The Talos star group was a star cluster in the Alpha Quadrant named after the Talosian planetary system. It was located in the third quadrant of the vernal galaxy between Rigel VII and the Vega colony, approximately six days away from Starbase 11 at maximum warp.
By the mid-23rd century, this cluster was unexplored and there were no Federation starships or colonies here. The only expedition into this region, a survey by the SS Columbia, had culminated in the Columbia disappearing under mysterious circumstances in 2236. Eighteen years later, in 2254, the USS Enterprise, under the command of Captain Christopher Pike, on a search-and-recovery mission for survivors of the downed survey ship became the second ship to enter this star group. (TOS: "The Cage", "The Menagerie, Part I")
Within thirty years of this mission, the Federation had expanded into this star group, and had established at least one colony in the Talos system. By the late-23rd century, the Talos star group was firmly within the Federation fold. (Star Trek II: The Wrath of Khan; Star Trek VI: The Undiscovered Country, production art)
The image of the Talos star group was a color astrophotographic exposure of M45, the Pleiades Cluster. The exposure was relatively short, and so was lacking detail of the characteristic Merope Nebula. However, the nebula does give a blue halo to the stars visible within the photograph.
The vernal galaxy information comes from the "For eyes of Starfleet Command only" file read by Kirk in "The Menagerie, Part I". It was unknown what vernal galaxy was, exactly, and how its quadrants are arranged. This reference system was never used again. Dialog in "The Cage" clearly states Earth and Talos to be in the same galaxy. According to Pike, the Enterprise originated from a stellar group at the other end of the galaxy relative to Talos.
"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".