The Alpha Quadrant.

The Alpha Quadrant was the common designation for one-quarter of the Milky Way Galaxy. It was adjacent to the Beta Quadrant and the Gamma Quadrant. One-quarter of the galactic core was located in this quadrant. (Star Trek: Voyager, Season 7 production art; Star Trek Into Darkness display graphics)

In late 24th century interstellar politics and diplomacy, the four great powers in the Alpha Quadrant were the United Federation of Planets, the Klingon Empire, the Romulan Star Empire, and the Cardassian Union. Other powers at the time were divided into two groups. The mercantile powers, like the Ferengi Alliance, traded with the great powers. The isolationist powers, like the Breen Confederacy and the Tholian Assembly, defended their territory aggressively and, would on occasion, battle with other powers. (Star Trek: The Original Series; Star Trek: Deep Space Nine; Star Trek Into Darkness display graphics) By the late 23rd century, the Alpha Quadrant still remained largely unexplored by the United Federation of Planets. (VOY: "Flashback")

The Bajoran wormhole connected the Bajoran sector in the Alpha Quadrant to a point near the Idran system on the far side of the Gamma Quadrant near Dominion space. The use of this wormhole for exploration and trade by parties from the Alpha Quadrant incited hostility from the Dominion, culminating in the Dominion War. (Star Trek: Deep Space Nine)

In the alternate 2259, a section of this quadrant, from a galactic map, was in a collection of graphics and video media that were seen on a powerwall in the offices of Admirals Christopher Pike and Alexander Marcus at Starfleet Headquarters. The map was being used primarily for charting activity on the Neutral Zone, Sectors 45 to 89. The three largest powers in this section - the Federation, the Empire, and the Orion Union - were labeled on this map. (Star Trek Into Darkness).

Background information.

In the script for DS9: "Playing God", the Cardassian vole was described as being "the Alpha Quadrant's nominee for ugliest creature in the galaxy".

In most reference works, apocryphal novels and games, Klingon and Romulan space and parts of the Federation are usually depicted or stated to be located in the Beta Quadrant. Though it may appear inconsistent with statements made from Star Trek: Deep Space Nine and Star Trek: Voyager, which place these powers in the Alpha Quadrant, it could be that the characters were talking about spheres of influence, whereby a power could exert its influence and prestige far from its homeworld. A good example of this is the Klingon Empire, with its homeworld in the Beta Quadrant, being called an Alpha Quadrant power. The Klingon Empire began exerting its influence and prestige in the Alpha Quadrant in the late 21st century when the empire expanded into the quadrant.

The Star Trek Encyclopedia states that the Federation was spread across the Alpha and Beta Quadrants in order to explain why the USS Enterprise was often the "only ship in the quadrant", though the term quadrant has been often used to refer to smaller segments of the galaxy as well.