STAR TREK QUADRANTS:

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 [1]; 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)

BETA QUADRANT

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 [1]; Star Trek Into Darkness, display graphics)
Qo'noS, the Klingon homeworld, was located in the Beta Quadrant. (Star Trek Into Darkness, display graphic) The Klingon Empire and the Romulan Star Empire claimed territory in the Beta Quadrant in the 24th century.

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 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")
A Borg ship, commandeered by liberated drones and Axum, was located in a remote sector of the Beta Quadrant patrolling the border of fluidic space. (VOY: "Unimatrix Zero, Part II")
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")

DELTA QUADRANT


The Delta Quadrant was the common designation for one-quarter of the Milky Way Galaxy. This quadrant was adjacent to the Beta Quadrant and to the Gamma Quadrant. The quadrant's closest point to the United Federation of Planets was located in the galactic core, which was located approximately 30,000 light years away. (Star Trek: Voyager, Season 7 production art [1])
The Delta Quadrant was home to thousands of species, including the Kazon, Talaxians, Ocampans, Borg, Vidiians, Hirogen, Krenim, and many others.
Humans were first planted in the Delta Quadrant by a race known as the Briori around the year 1937 as a form of slave labor. The slaves rebelled and their descendants built a new civilization on a Class L planet. (VOY: "The 37's")
The Delta Quadrant was first visited by Humans of their own free will on stardate 32629.4, when the USS Raven followed a Borg vessel into a transwarp conduit. (VOY: "Dark Frontier") The first Starfleet mission into the Delta Quadrant occurred during an inspection of the Barzan wormhole in 2366. (TNG: "The Price")
Due to the immense distance between the Alpha and Delta Quadrants (an Intrepid-class starship would take thirty years to reach the edges of the quadrant at its maximum warp velocity), the Federation knew very little about this region of space until 2371, when the starship USS Voyager was pulled into the quadrant by an alien force called the Caretaker. When contact was reestablished with the starship in 2374, hundreds of kiloquads of data on the region was received, increasing the Federation's knowledge of the quadrant immensely.
When Q discovered Voyager, he assumed Quinn was responsible for transporting it there, stating that Humans weren't supposed to be in the Delta Quadrant for another one hundred years. (VOY: "Death Wish")
When Voyager returned to Earth from the Borg transwarp network in 2378, the Federation's knowledge of the quadrant was expanded considerably. (VOY: "Endgame")


Millions of years before recorded Human history, it is speculated that the Voth may have left Earth and traveled to the Delta Quadrant where they established a powerful and reclusive civilization. (VOY: "Distant Origin")
Around 45,000 years ago, a humanoid species from the Delta Quadrant traveled to Earth, where they discovered a tribe of Humans. Though the Humans at this time had no language or tools, the humanoid visitors admired their respect for the land. The humanoids continued to visit Earth for thousands more years, seeing the tribe they visited become the Native American people, and they came to worship the humanoids as the Sky Spirits. The humanoids stopped visiting when European settlers started to conquer their land. (VOY: "Tattoo")
Around the 3rd century, an epidemic known as Phage spread throughout the Vidiian Sodality, reducing a previously renowned culture to subsisting on organ piracy and slave labor. (VOY: "Phage", "Faces")
In the 15th century, the Vaadwaur controlled many colonies and exerted influence across much of the Delta Quadrant, if not further, via a series of underspace corridors. In 1484 an alliance of species led by the Turei destroyed the Vaadwaur empire, forcing survivors to undergo cryostasis in the hopes of preserving their society. (VOY: "Dragon's Teeth")
Jal Sankur united the Kazon sects in 2346 to overthrow their enslavers, the Trabe. Virtually all Trabe territory fell under control of the various sects, which soon began fighting amongst each other. (VOY: "Initiations", "Maneuvers", "Alliances")
In 2369, Q told Vash that there was still the Delta Quadrant for her to explore with him. (DS9: "Q-Less")
Prior to 2371, an entity known as the Caretaker, while not claiming any specific territory per se, was a major factor in the sociopolitical climate near the outer rim of the Delta Quadrant, in the midst of Kazon space. (VOY: "Caretaker")

The great power in the Delta Quadrant, until at least 2378, was the Borg Collective. In addition to a vast swath of territory which they had completely assimilated, the Borg used a transwarp network to routinely visit different points scattered throughout the quadrant. Following the Borg-Species 8472 War, the insurrection led by Axum, the destruction of a part of the transwarp network and the destruction of Unimatrix 01, it was unclear how the Borg's status in the Delta Quadrant had been altered. (VOY: "Scorpion", "Unimatrix Zero", "Endgame")
Other alien species that controlled significant areas of the quadrant included, among others, the Vidiians, the Devore, the Kazon, the Voth, the Hirogen, the Hierarchy, and the Malon. Species 8472 had also established a presence in the Delta Quadrant until they returned to fluidic space. (Star Trek: Voyager)
The political power of the Krenim Imperium depended largely on its temporal weapons, and as such its influence may have been altered dramatically by its own manipulation of the timeline. (VOY: "Year of Hell")

GAMMA QUADRANT

 The Gamma Quadrant was the common designation for one-quarter of the Milky Way Galaxy. This quadrant was adjacent to the Alpha Quadrant and to the Delta Quadrant. One-quarter of the galactic core was located in this quadrant. The quadrant's closest point to the United Federation of Planets was located in the galactic core, which was located approximately 30,000 light years away. The Denkiri Arm was located in this quadrant. (TNG: "The Price"; Star Trek: Voyager, Season 7 production art 

Vash commented that some of the cultures she had encountered in the Gamma Quadrant had histories spanning millions of years. (DS9: "Q-Less") It is believed the Hur'q originated from the quadrant. (DS9: "The Sword of Kahless")
Using their system of gateways, the Iconians were able to settle the world of Vandros IV, on the edge of the Gamma Quadrant. (DS9: "To the Death")

The quadrant was first explored by the Federation's Quadros-1 probe in the 22nd century. (DS9: "Emissary")
A Barzan probe sent through the Barzan wormhole discovered that the Gamma Quadrant terminus was located beyond the Denkiri Arm, a distance that at warp 9 would have taken nearly a century to cover, this later shifted to the Delta Quadrant and the wormhole was discovered to be unstable. (TNG: "The Price")
In 2369, the Bajoran wormhole in the Bajoran system was discovered. This wormhole had a terminus in the Gamma Quadrant that was four light years from the Idran system. This wormhole linked this quadrant with the Alpha Quadrant. (DS9: "Emissary")
A Tosk came through the Bajoran Wormhole from the Gamma Quadrant and docked at Deep Space 9 on a hunt and was pursed by Hunters, The Hunters said that Tosk was bred to be hunted by them and had spent his entire life preparing, later the Hunters were allowed to resume the hunt and take the Tosk back with them to the Gamma Quadrant after Chief O'Brien helped the Tosk to escape and leave back through the Bajoran Wormhole and into the Gamma Quadrant. (DS9: "Captive Pursuit")
The archaeologist Vash explored the Gamma Quadrant for two years at the invitation of Q in 2367. She was found there in 2369 by members of Deep Space 9's crew. (TNG: "Qpid"; DS9: "Q-Less")
Deep Space Nine had the distinction of making first contact with a Gamma Quadrant species known as the Wadi. However, the Wadi showed no interest in formalities and headed straight to Quark's. The Ferengi initially saw the Wadi as a potential source of great profit, but when the Wadi Master Surchid Falow started to win at Dabo and went on a winning streak, Quark had one of his waiters rig the game. The Wadi caught him and forced Quark to play "an honest game" called Chula, using living people as players, but this was revealed later to be only a game and not really dangerous. (DS9: "Move Along Home")
Kai Opaka was taken through the Bajoran Wormhole by Benjamin Sisko, where they received a distress signal. They then found a planet with a network of satellites, one of which was malfunctioning. Their runabout approached the planet and one of the satellites attacked it, forcing them to crash land. Opaka died in the landing, which devastated Kira; however, her mourning was interrupted by a group of people called the Ennis are who were at war with a group called the Nol-Ennis, both groups were being regenerated by some form of extraterrestrial nanotechnology. The same technology revives the Ennis and Nol-Ennis, but rather than a blessing it is a curse: Both groups were sent to the planet as punishment for centuries of conflict, doomed to die over and over, Kai Opaka accepts that she cannot leave the planet in the Gamma Quadrant and feels called to help the two warring factions learn how to live in peace. (DS9: "Battle Lines")
The Vulcans were at the forefront of the exploration of the Quadrant, beginning only months after the wormhole was discovered by Benjamin Sisko and Jadzia Dax in 2369, the Vulcans made many discoveries, including the remains of the Hur'q race. (DS9: "The Sword of Kahless")
The Klingons showed an interest in exploring the Gamma Quadrant shortly after its discovery by sending their own scout ships to the quadrant. (DS9: "Q-Less") The IKS Toh'Kaht was among the first ships to explore the quadrant in an effort to establish colonies on the other side of the wormhole. They also operated freighters to the quadrant. (DS9: "Dramatis Personae", "The Storyteller")
On the other hand, no Romulan ship had ever been to the Gamma Quadrant until late 2371 during the Tal Shiar's abortive attack on the Founders' homeworld. (DS9: "Visionary", "The Die is Cast")
The Ferengi saw the Gamma Quadrant as an enormous chance to make a profit and their reputations would be, as Gral put "absolutely stainless". (DS9: "The Nagus", "Rules of Acquisition", "Starship Down")
The Bajorans also colonized a planet in the Gamma Quadrant, New Bajor, in late 2370. Unfortunately, the planet was invaded by the Jem'Hadar, who massacred the population. (DS9: "Crossover", "The Jem'Hadar")
The planet Meridian was also located in the Gamma Quadrant, and spends most of its time in another phase of existence, where its people exist only as consciousness, with no physical being. The planet is in this dimension for a very short time. In twelve days it will disappear for sixty years. Each time it returns, the planet will have less time in this universe, and eventually will stop appearing. (DS9: "Meridian")
The starship Defiant was caught in a type of "quantum-bubble" surrounding an inhabited planet. After investigation, it is discovered that the inhabitants are the descendants of the Defiant's crew. Generations of descendants have been living on the planet for many years and their own history began when the Defiant tried to escape from the bubble. Instead of passing through its energy barrier, the ship was thrown back in time two centuries and crashed with no hope of contacting the Federation or repairing the ship. Faced with that reality, the small stranded crew of the Defiant (including Dax, Sisko, O'Brien, etc.) decided to remain and establish a society. (DS9: "Children of Time")

The Dominion was established approximately 10,000 years prior to 2375. This galactic power was located between 40,000 and 50,000 light years from the galactic core in an arm of the galaxy. Other powers have been encountered, but their status with the Dominion was unclear. (DS9: "The Dogs of War", "Whispers", "Hard Time"; Star Trek: Voyager, Season 7 production art)
In the act directly precipitous to the Dominion War, the Federation mined the entrance to the wormhole (leading to the Gamma Quadrant) in 2373. This cut off access to this quadrant from any Alpha Quadrant powers and prevented Dominion reinforcements from adding to the Dominion War. (DS9: "Call to Arms", "Sacrifice of Angels", "Tears of the Prophets")