(covers information from several alternate timelines)
Augment is a term used to describe a group of genetically engineered Humans created by advances in DNA resequencing in the mid 20th century. In the 22nd century, Klingons created their own Augments using the original augmented Human DNA.
The Augments were designed to be remarkably agile, five times as strong and twice as intelligent as a normal Human, resistant to sickness and with enhanced senses, possessing heart muscles twice as strong and lung efficiency 50% better. Their blood contained platelets capable of regenerating from any disease or toxin, which could be used to cure or revive medical subjects via transfusion. They also had twice the average lifespan. Even their resistance to directed energy weapons was improved, as it took multiple shots with a phaser or a phase pistol to stun one. They were even capable of resisting a Vulcan nerve pinch and mind meld. (TOS: "Space Seed"; ENT: "Borderland", "Cold Station 12"; Star Trek Into Darkness)
The intelligence of an Augment was exemplified by how, within a year of being awakened, Khan Noonien Singh was already capable of designing entire advanced systems of mass destruction, including the USS Vengeance, its advanced beam and missile weaponry, a propulsion system that enables the interception of a ship while in warp, and torpedoes capable of eradicating large portions of a planet. He was able to do this, despite being armed with outdated 300 year old technical knowledge before being put in stasis. The ability to learn all of the latest technologies, beyond many post-doctoral qualifications, is a testament to how their artificially augmented superhuman intelligence was beyond the reach of even the most intelligent humans. (Star Trek Into Darkness)
Along with their superior abilities there was a defect in their genome: the Augments were aggressive, arrogant and ambitious, with a diminished sense of morality. One of the scientists behind their creation said that "superior ability breeds superior ambition." Doctor Arik Soong later theorized that a defect in the genomes of the Augments created a malformation in the base-pair sequences that regulate the neurotransmitter levels in their brains, causing them to be highly prone to aggression and violent behavior, and considered fixing this defect before incubating some embryos. (ENT: "The Augments")
The Eugenics Wars
In 1992, the Augments born around the late 1950s seized power in over forty nations on Earth, leading to the conflict known to them as the Great Wars, as they battled amongst each other and with "normal" Humans. The Augments were mostly horrible despots, treating their subjects like slaves. (TOS: "Space Seed"; Star Trek Into Darkness)
The Augments were finally overthrown in 1996; the last to be deposed was Khan Noonien Singh. After the end of the war some eighty Augments were unaccounted for, including Khan. (Star Trek II: The Wrath of Khan) Earth authorities collected the remaining Augment embryos and placed them into storage; by the 2130s they were stored on the Earth Starfleet facility Cold Station 12. (ENT: "Cold Station 12") As a result of the Wars, genetic engineering was banned on Earth; scientist Stavos Keniclius who planned to clone Augments to create a master race of peace keepers, was exiled from the planet. (TAS: "The Infinite Vulcan")
In the 2130s, Arik Soong, senior medical director of Cold Station 12, managed to steal several Augment embryos from the facility, which he took to the Trialas system, where he raised the resulting children as his own. A decade later he was captured by Earth forces, but refused to disclose what had happened to the embryos. (ENT: "Cold Station 12")
In May 2154 the answer to this puzzle was discovered when Soong's Augments, led by Malik, seized control of a Klingon Bird-of-Prey, jettisoning its crew into space. (ENT: "Borderland") The Klingons recovered these bodies, and discovering the Human DNA on them, threatened war. The NX-01 Enterprise was launched to the Borderland to find these Augments and bring them back to Earth, and Arik Soong was recruited to assist the ship in doing so.
Unfortunately, Soong managed to escape on Verex III and rejoin his Augments, who then planned to use their bird-of-prey to free the thousands of other embryos on Cold Station 12. (ENT: "Borderland") All of these Augments were killed when Malik destroyed their bird-of-prey to avoid capture. Soong, who "defected" to Enterprise, decided afterwards to switch his focus to cybernetics. (ENT: "The Augments")
See also: The Augment Crisis
After the Augment crisis, the Klingons attempted to create their own augments to combat the Humans, believing that Humans would place Augments on their ships and put the Klingons at a severe disadvantage, despite Vulcan's assurance that Earth had banned such genetic enhancement decades ago. They were able to retrieve multiple embryos left over in the debris of the Augment Bird-of-Prey and implemented their DNA in certain Klingons. However, due to the aggressiveness of Augment DNA, the cranial ridges started to dissolve and the Klingon Augments completely lacked ridges on their foreheads.
These subjects did gain increased strength and intelligence, but they died when their neural pathways degraded. Unfortunately, one of the subjects was suffering from Levodian flu, which was modified by the Augment genes to become a deadly, airborne plague that spread throughout the Empire. An early symptom of this plague caused victims to lose the ridges on their foreheads and to look more Human. Other symptoms included irregular heartbeat, chest pain, and tingling in the cranial ridges.
A cure was eventually created by Phlox which halted the virus in the first stage, retaining the changes in appearance but with no enhanced strength, speed, or endurance. This left millions of Klingons without their ridges, an alteration that was even passed on to their children, though it was hoped that one day gene therapy would be developed to reverse the effects (ENT: "Affliction", "Divergence"; DS9: "Trials and Tribble-ations"; TOS et al.). The effects of this virus were reversed in at least part of the population by 2273. (Star Trek: The Motion Picture)
The Return of Khan
The fate of Khan and his Augments remained unknown until 2267, when the USS Enterprise discovered the SS Botany Bay, a DY-100 class sleeper ship. Aboard the ship was Khan and 72 of his fellow Augments (twelve had died due to stasis unit failure in the interim). At first the Enterprise crew was unaware of the true identity of Khan, but when it was discovered, Khan attempted to take the ship, though he ultimately failed.
Enterprise commanding officer James T. Kirk generously offered Khan and his followers voluntary exile to the uninhabited Class M planet Ceti Alpha V, and Khan accepted. (TOS: "Space Seed") Unfortunately, six months later the nearby planet Ceti Alpha VI exploded, causing the orbit of Ceti Alpha V to shift, which devastated the planet's environment. The dominant remaining lifeform was the Ceti eel, which claimed many of Khan's followers.
In 2285, Khan seized control of the USS Reliant, intending to use it to destroy Kirk in revenge for what had happened to him and his people. In the end, Khan died when Reliant was destroyed by the detonation of the Genesis Device after a heated battle in the Mutara Nebula. (Star Trek II: The Wrath of Khan)
In 2258 of the alternate reality, Admiral Alexander Marcus revived Khan Noonien Singh and using his crew as leverage, recruited him into Section 31. Khan served Marcus under the identity of John Harrison. Marcus believed Khan's savage brilliance was required for an inevitable conflict with the Klingon Empire. In 2259, after Khan thought Marcus had executed his crew, he went rogue. Khan orchestrated a bombing in London and attacked Starfleet Headquarters. He eventually murdered Marcus and crashed the severely damaged USS Vengeance into San Francisco. Spock and Uhura apprehended Khan and he was placed back into suspended animation. (Star Trek Into Darkness)
The term "Augment" itself was only used in the 22nd century episodes of ENT. The term was only known to have applied to those whose DNA was genetically engineered in the 20th century. The following episodes featured Human Augments: