The Q Continuum was an extra-dimensional plane of existence inhabited by a race of beings known as the Q. The term could also refer to the Q society itself.
As a race, the Q were immortal, but not absolutely omniscient or omnipotent (according to Quinn). They possessed the ability to alter, create, or destroy virtually any form of matter or energy, and to affect time in essentially any way they chose. Their apparent abilities included moving entire asteroid belts and stars, creating alternate timelines, and affecting universal states of nature such as the gravitational constant. Q, when discussing the dangers of allowing Amanda Rogers to develop her abilities without supervision, offhandedly remarked that she would be capable of destroying the entire Milky Way Galaxy by accident. In fact, the only time a Q was seen as unable to do something was during a period where their powers were reduced or revoked by the Continuum. (TNG: "Deja Q", "Tapestry", "All Good Things...")
Based on descriptions by numerous Q, including the Q later known as Quinn, as well as Amanda Rogers (the child of two Q in Human form), the Q Continuum was a very highly ordered society, but also a virtually stagnant one.
In the beginnings of what Quinn called the "new era", the Q engaged in constant dialogues of discovery and other universal issues. However, eventually, the Q came to realize that they had experienced, observed, or discussed literally everything imaginable. This resulted in the virtual shutdown of the Continuum as a society, as the immortal beings simply saw no point in even conversing with one another anymore. This resulted in one of the Q, Q, deciding to simply use his powers to stir up trouble and cause chaos, simply because there was nothing else to do. As a result of Q's rebellious behavior, Quinn, a respected philosopher in the Continuum, decided that he wanted to shake things up even further by committing suicide, as he not only saw no purpose in continuing to exist, but wanted to break the monotonous stagnation afflicting the Continuum. The Continuum, however, stopped him from doing so because they were unable to predict what sort of societal chaos might result in the Continuum as a result of a Q committing suicide.
Quinn was imprisoned on a rogue comet for eternity to prevent him from ending his own life, until he was accidentally released by the crew of the USS Voyager in 2372. In a courtroom hearing to determine his right to political asylum, Quinn described the Continuum as a lazy, old way station along a deserted desert road – he argued that the road could take them to anywhere in the universe, but the Q had already been everywhere. In addition, everything both old and new had already been discussed, and so the roadhouse was silent – there was nothing left to say. Therefore, Quinn argued that being forced to remain alive was a continued burden to him, a burden that he did not want to continue to bear. Q argued that the death of a Q would create chaos in the Continuum. Voyager captain Kathryn Janeway ruled in Quinn's favor, granting him asylum. A day later, Quinn committed suicide. (VOY: "Death Wish")
As a result of Quinn's action, the Continuum was plunged into a massively destructive civil war that set the "Freedom Faction" against the traditional Q. The two factions were able to construct weapons that could even compromise the immortality of a Q. When these weapons were fired in the Continuum, this resulted in massive damage to subspace, causing some stars in normal space to go supernova. Eventually it damaged subspace to such an extent that Q outside of the Continuum lost their powers and immortality.
Q had the idea of mating with Janeway to produce a being with the powers of a Q, and to put the morals of a Human to use in order to end the war. The crew of Voyager flew through a supernova into the Continuum with help from a female Q, who had lost her powers and could not return without Voyager's help. Voyager's crew quickly overpowered the Q by using Q weapons. A ceasefire was called. Q mated with the female Q instead of Janeway and the damage to subspace was restored. (VOY: "The Q and the Grey")
Later on, the new Q became an intergalactic troublemaker. He started wars between innocent races, knocked planets out of orbit, detonated Omega molecules, and created havoc in every way he could. His mother disowned him, humiliated by his actions. Q, however, followed him, cleaning up all the damage.
Under advice from Janeway, Q implemented punishment on the boy. Q dumped him on Voyager, and, with the help of the "Q government", took away his powers. Q Junior, as he was called by his father, had one week to become a good citizen, or he would be sentenced to eternity as an Oprelian amoeba. Junior did well for the first few days, until he stole the Delta Flyer. After his friend Icheb was injured, however, he returned to Voyager. Janeway made him apologize to the ship he attacked, which was in fact Q, who masqueraded as a likely target to test his son's attitude.
When Judgment Day came, Junior was still found unworthy of being a Q, but was good enough to be a Human. After this sentence, Q threatened to leave the Continuum unless his son was allowed to stay. After further pleading, the Continuum agreed to grant the young Q's powers back to him on the condition that his father act as his guardian and supervisor in perpetuity, or at least until he could prove himself worthy. (VOY: "Q2")
The history of the Q Continuum is explored in the Q Continuum series, which features flashbacks to Q's "youth", including him being sentenced to safeguard Earth as punishment after his actions released a powerful entity known as "0" into the universe when Earth was damaged in the last stage of the war with 0 and his allies, which included Gorgan and the Beta XII-A entity. The conclusion of the trilogy sees 0 briefly return to this universe, achieving a level of power temporarily beyond even Q himself as his insanity from millennia of solitude allows him to warp reality at a fundamental level that Q would never be able to do without risking damage to himself. The series revealed that the Continuum was responsible for the creation of the Galactic barrier and for the imprisonment of the entity they knew as The One on the planet dubbed "Sha Ka Ree" by Sybok. This barrier was weakened as a result of the civil war in the Continuum, which was what allowed 0 to make his return, but later repaired after 0 was exiled yet again.
In The Eternal Tide, it is revealed that the Q Continuum manifested as a creative force to "oppose" the destructive potential of the Omega Continuum after it was disrupted and the decay of the universe accelerated. Voyager is later able to slow down the decay to a more tolerable level – shortening the universe's lifespan by a few million years rather than by trillions – as the only way to completely halt and undo the disruption would have erased the Q Continuum from existence as their creative force was no longer "needed". Eternal Tide also reveals that there are at least six rules that the Q must obey, with number six being "Don't bring the dead back to life", as the consequences of such an action can be problematic as not even the Q fully know what happens after death.
In the Star Trek: Ongoing story arc The Q Gambit, it is revealed that the higher species are at war and the Q Continuum is on the verge of defeat. Galvanized by their victory over the Prophets, the Pah-wraiths have turned on their other neighbors. Not even the Q cannot stop their onslaught because in spite of all the power the Q wield in the three-dimensional universe, they are as powerless and clueless as humans in their own realm. Unable to find a path to victory, Q left to seek the counsel of Picard on what action to take. But when he couldn't get an answer, Q left for the alternate reality, intended to instead seek the counsel of James T. Kirk for his experience in triumphing over no-win scenarios.