The First Federation was a political and military organization in the Alpha Quadrant. The location of this organization was depicted on a Federation star chart in 2293. (TOS: "The Corbomite Maneuver"; Star Trek VI: The Undiscovered Country)
First contact between the Federation and the First Federation occurred in 2266, when the USS Enterprise was conducting a star mapping mission and unknowingly crossed into First Federation space, destroying a marker buoy in the process.
Shortly thereafter, the Enterprise encountered the massive flagship of the First Federation, the Fesarius. Charged with violating their territory, the Enterprise and its crew were threatened with destruction. However, after a series of bluffs and counter-bluffs, it was later discovered that the encounter had been a test, designed by the Fesarius' captain, Balok. A cultural exchange was proposed, with Enterprise crewman Dave Bailey remaining on board the Fesarius. (TOS: "The Corbomite Maneuver")
"The Explored Galaxy"
"The Explored Galaxy" was a map of charted space in the Alpha Quadrant. The First Federation was located in an area of space that was located west of the Tholian Assembly. The Assembly was identified in DS9: "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 First Federation was located in the Alpha Quadrant. The First Federation's capital system was Fesarius. Fesarius was a Class K star with a magnitude of +3, which was ten times the brightness of Sol. A major space lane was located in proximity to the territory of the First Federation. Warning buoys had been installed by 2378 around the First Federation's territory. (Pg. 36, "United Federation of Planets I")