Beta Quadrant
The Beta Quadrant would be located between the 3 and 6 o'clock if the plane of the galaxy were seen as a clock face with the 6 o'clock position bisecting Earth's Solar System.
The reason why the United Federation of Planets and Earth is right on the edge of the Beta Quadrant is to make previous references to Quadrant made in the original series, its movies, and early episodes of Star Trek: The Next Generation consistent.
The Beta Quadrant is home to the Klingon Empire, Romulan Star Empire, Gorn Republic, Son'a and Metron Consortium as well as a portion of the Federation, including Vulcan. Parts of the Romulan and Klingon empires spill into the Alpha Quadrant, and thus the Romulans and Klingons are still referred to as Alpha Quadrant Powers.
See also: Alpha Quadrant, Gamma Quadrant, Delta Quadrant.