The Beta Quadrant was the common designation during the 24th Century for one-quarter of the Milky Way Galaxy when viewed from a point of view perpendicular to the plane of the galaxy. It extended ninety degrees spinward of an imaginary line from the center of the galaxy through Sector 001 (which contains the Sol System.)
The Carina Arm and Norma Arms of the galaxy are located in that quadrant, as is the star NGC 5139.
The majority portions of the Klingon Empire and the Romulan Star Empire were located in the Beta Quadrant. The Federation also claimed territory in that Quadrant in the 23rd and 24th centuries, although it was more frequently identified during that time as an Alpha Quadrant power.
The Yattho, a species known for the precognitive abilities, were native to the Beta Quadrant.
The Federation knew little of those portions of the Beta Quadrant not already claimed by known powers well into the 24th Century.
In 2293 the starship USS Excelsior, commanded by Captain Sulu, completed a three year exploratory voyage in the Beta Quadrant which included cataloguing gaseous anomalies. Seventy years later Captain Lisa Cusak and the crew of the USS Olympia began a seven year exploration voyage in the Beta Quadrant.
A Borg ship, comandeered by liberated drones and Axum, was located in a remote sector of the Beta Quadrant.
See also
From Memory Alpha [1] -- the free Star Trek reference