The Alpha Quadrant is a term designating a quarter of the Milky Way Galaxy, encompassing parts of the United Federation of Planets, including Earth, Betazed, Delta IV, and Trill, and other powers, such as the Cardassian Union, the Ferengi Alliance, the Tzenkethi, the Tholian Assembly, the Xindi and the Breen Confederacy.
The Sol system rests on the border between the Alpha and Beta Quadrants but is considered in Sector 001 of the Alpha Quadrant. Major galactic arms included in the Alpha Quadrant are the Orion Arm, the Perseus Arm and the Sagittarius Arm, while major astronomical landmarks included the Argolis Cluster, the Arachnid Nebula and the Badlands. (ENT: "Fusion"; TNG: "I, Borg"; DS9: "The Maquis, Part I")
In the 24th century, Federation scientists approximated all known races had explored only 25% of the quadrant. During these explorations, three stable wormholes were discovered: the Barzan wormhole, leading to the Delta Quadrant (which was ultimately discovered to be unstable at the Delta Quadrant end); and the artificial Bajoran wormhole, reaching 70,000 light years into the Gamma Quadrant. (TNG: "The Price"; VOY: "False Profits"; DS9: "Emissary"))
The Dominion infiltrated the Alpha Quadrant through the Bajoran wormhole, finally invading in 2373, causing the Dominion War, lasting until 2375, when the major Alpha Quadrant powers drove the Dominion to surrender. (Star Trek: Deep Space Nine)