(covers information from several alternate timelines)
|The quadrants of the Milky Way|
| The Beta Quadrant is located in the lower left|
(click to navigate)
The Beta Quadrant was the common designation for one-quarter of the Milky Way Galaxy. It was adjacent to the Alpha and Delta Quadrants. One-quarter of the galactic core was located in this quadrant. (Star Trek: Voyager, Season 7 production art;  Star Trek Into Darkness display graphics)
Part of the Klingon Empire, including Qo'noS, was located in the Beta Quadrant. (Star Trek Into Darkness display graphic; DIS: "Context Is for Kings" display graphic) The Romulan Star Empire claimed territory in the Beta Quadrant in the 24th century, including their adoptive homeworld, Romulus. (Star Trek: Insurrection display graphic; PIC: "The End is the Beginning")
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 an eight-year mission 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 an alternate timeline, the USS Voyager used a quantum slipstream drive to cross the Delta and Beta Quadrants, before crashing on a 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")
Captain Riker – holoprogram
In 2367, while exploring a cavern on Alpha Onias III, Commander William T. Riker was rendered unconscious by gases. While unconscious, neural scanners scanned Commander Riker's brain. The scanners used elements of Riker's reality and constructed a holoprogram with those elements interspersed throughout, so that it felt real to him. What Riker wanted, the scanners made possible.
In one of these simulations, the Axanar hosted a conference on Beta Quadrant exploration and Federation security that was held at the Deneva Special Congress on Interstellar Affairs. On stardate 52384, Captain Riker, the commanding officer of the USS Enterprise-D, was selected for recognition as the special Federation representative to this conference. In 2383, the quadrant was mentioned in Riker's service record that was on display in his quarters on the Enterprise-D. (TNG: "Future Imperfect" okudagram)
Alpha/Beta quadrant border
The Star Trek Encyclopedia (2nd ed., p. 393) states that the Federation was spread across the Alpha and Beta Quadrants, with Earth as the border between, 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. The maps in the Star Trek: Star Charts also use Earth as the border between the Alpha and Beta quadrants.
Graphics used in astrometrics for Star Trek: Voyager, starting with the episode "Year of Hell", and on a PADD on Picard's ready room desk in Star Trek: Insurrection, also have Earth and the Federation straddling the border between the two quadrants. 
Dialogue in Star Trek: Deep Space Nine and Star Trek: Voyager, however, firmly establish that Earth is located on the Alpha Quadrant side of the border. Graphics used in Star Trek: Discovery and the alternate reality films support this, where the Alpha/Beta Quadrant border is moved one sector "west" and a star chart graphic used in Star Trek Into Darkness had the label "Alpha Quadrant" near Sector 01 (Level 3-B), which contains Earth. 
According to the Star Trek Encyclopedia (2nd ed., p. 393), Federation, Klingon, and Romulan territories are situated in both the Alpha and Beta quadrants. Dialogue from Deep Space Nine and Voyager exclusively referred to all three powers and governments in their vicinity as Alpha Quadrant powers.
In most reference works, novels, and games, Klingon and Romulan space are usually depicted or stated to be located solely in the Beta Quadrant. This would make the Federation the only government spanning both quadrants.
The video game Star Trek Online is largely faithful to the Star Charts representation of the galaxy, with a few systems moved for gameplay considerations.
According to the Star Charts, most of the early voyages of Enterprise NX-01 took place in the Beta Quadrant with only a few excursions in the Alpha Quadrant.
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 ship's 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: Star Charts, the Beta Quadrant was home to the Vulcans, Andorians, Orions, Bolians, Son'a, Risians, Organians, Coridanite, Axanar, Nausicaans, Suliban, Gorn, and many other alien races.