(covers information from several alternate timelines)
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 drawn from the center of the galaxy, the galactic core, passing at least near the Sol system. (VOY: "Pathfinder", "Year of Hell")
In 2293, the starship USS Excelsior, commanded by Captain Hikaru Sulu, completed a three-year exploratory voyage in the Beta Quadrant which included cataloging gaseous anomalies. (Star Trek VI: The Undiscovered Country)
In 2363, Captain Lisa Cusak and the crew of the USS Olympia began a seven-year exploration voyage in the Beta Quadrant. Following this mission, the Olympia crashed on a planet in the Rutharian sector en route back to Federation space. While escorting a convoy to the Vegan system in 2374, the crew of the USS Defiant made contact with Cusak in 2371 via a temporal anomaly. (DS9: "The Sound of Her Voice")
In 2367, Barash created a holoprogram of a fictional future based on William T. Riker's thoughts and expectations. In Riker's fictional service record in the holoprogram, on stardate 52384, he was selected for recognition as the Federation special representative to the Axanar conference on Beta Quadrant exploration and Federation security that was held at the Deneva Special Congress on Interstellar Affairs. (TNG: "Future Imperfect")
In an alternate timeline, the USS Voyager used a quantum slipstream drive to cross the Delta and Beta Quadrants, before crashing on an Class L planet in the Takara sector, a few parsecs short of reaching the Alpha Quadrant. Also in this timeline, Starfleet Intelligence had salvaged a Borg temporal transmitter from the debris of a Borg cube in the Beta Quadrant. (VOY: "Timeless")
When members of the Hierarchy captured Captain Janeway and forced The Doctor to obtain Voyager's warp core for them, he impersonated Janeway and told Commander Chakotay a cover story involving a fictitious R'Kaal Imperium which supposedly controlled "thousands of parsecs from here to the edge of the Beta Quadrant." (VOY: "Renaissance Man")
The Fen Domar, with whom Voyager was engaged in battle in an alternate timeline, may very likely be a Beta Quadrant civilization. This is based on the ships close proximity to the quadrant's border in "Endgame" and Admiral Kathryn Janeway's comments that the ship would have encountered them just a few years after the events of that episode.
According to Star Trek Encyclopedia, Earth marks the border between Alpha and Beta Quadrants. This notion seems to be supported by a graphic from "Year of Hell, Part I" and the display graphic on the PADD from Star Trek: Insurrection.  Dialogue in Star Trek: Voyager and Star Trek: Deep Space Nine however firmly establish that Earth is located on the Alpha Quadrant side of the border. According to Star Trek Encyclopedia, Federation, Klingon, and Romulan territories are situated both in the Alpha and Beta Quadrants. Again, in dialogue from Star Trek: Voyager and Star Trek: Deep Space Nine all three powers and governments in their vicinities have been exclusively referred to as being located in the Alpha Quadrant.
According to Star Trek: Star Charts, the Beta Quadrant was home to the Vulcans, Andorians, Orions, Bolians, Son'a, Risians, Organians, Coridan, Axanar, Nausicaans, Suliban, Gorn and a many other alien races. According to the book, some of the early voyages of the Enterprise NX-01 took place in the Beta Quadrant with only a few excursions into the Alpha Quadrant.