This page contains information regarding new Star Trek material, and thus may contain spoilers.
"Nero's very presence has altered the flow of history, beginning with the attack on the USS Kelvin, culminating in the events of today, thereby creating an entire new chain of incidents that cannot be anticipated by either party."
"An alternate reality?"
"Precisely. Whatever our lives might have been, if the time continuum was disrupted, our destinies have changed."
The alternate reality was the new reality created when the temporal incursion of the Narada from the year 2387 disrupted the time continuum. Accidentally traveling back to the year 2233, Nero, the captain of the Narada, attacked the USS Kelvin. The attack resulted in the deaths of George Kirk and Richard Robau and the destruction of the Kelvin. Spock arrived to the alternate reality in 2258 and was captured by Nero, who used red matter to destroy Vulcan. However, Nero's attacks united the crew of the USS Enterprise years earlier, to foil his attempt to destroy Earth. (Star Trek)
In the prime universe, the supernova of 2387 threatened the entire galaxy. Ambassador Spock was able to halt the supernova, via the use of red matter to create an artificial singularity, or black hole, which absorbed the exploding star, but was too late to save the planet Romulus from destruction. The Narada, a Romulan mining ship under the command of Captain Nero, was pulled into the black hole, followed by Spock's ship, the Jellyfish.
Nero placed blame on the Federation for the loss of his homeworld and sought revenge. He emerged from the black hole in 2233. The USS Kelvin was the first ship that Nero encountered and attacked; Captain Richard Robau promoted his first officer, Lieutenant Commander George Kirk, to captaincy before ordering the evacuation of the ship and agreeing to come aboard the Narada. There, Captain Robau was interrogated regarding the whereabouts of Spock, with whom Robau was unfamiliar and, moments after he informed Nero of the current stardate, he was murdered.
Nero then proceeded to attack the Kelvin. Kirk used the Kelvin's weapons to prevent Nero from destroying the evacuating shuttles departing the ship, ultimately sacrificing himself by ramming the Kelvin into the Narada. Kirk's actions saved some 800 lives, including his wife, Winona Kirk, and their newborn son, James, but failed to destroy the Narada.
As a major consequence of these events, James Kirk grew up without his father and without the ambitions his father gave him in the prime reality. However, he was persuaded by Christopher Pike to join Starfleet, five years later than he had done in the prime reality.
In the meantime, other events happened differently. Pavel Chekov was born in 2241, while the Romulans were confirmed as relatives of the Vulcans. Plans for the Constitution class were pushed back by a decade and the USS Enterprise began construction in 2255 at the Riverside Shipyard in Iowa, with a number of internal, external and systems design differences to the prime reality version. It was launched three years later, as the Federation flagship under the command of captain Pike. Spock was already promoted to Commander by this point. Starfleet continued using the simple <earth calendar year>.<day of the year> format for stardates, and began using gold, blue and red colors for their uniforms. They also adopted the Kelvin assignment patch as the sole Starfleet insignia. The Vulcan High Command was reinstated as the Vulcan High Council.
Destruction of Vulcan
In 2258, Nero captured the Jellyfish as it emerged from travel through the black hole. He marooned Spock on Delta Vega, and despite the efforts of the USS Enterprise, he used a portion of the remaining red matter aboard to destroy the planet Vulcan and six billion of its inhabitants, including Amanda Grayson. While marooned on Delta Vega, Spock encountered the alternate James T. Kirk and made him aware of the prime reality and the altered past. He also met Montgomery Scott and gave him the formula for transwarp beaming, which Scott subsequently used to transport Kirk and the young Spock to the Narada, allowing them to prevent a similar fate for Earth by detonating the Jellyfish and its red matter to destroy the Narada.
Afterward, Christopher Pike was promoted to Admiral and Kirk and Spock became captain and first officer of the Enterprise, which was officially launched on a voyage of exploration, with much of its commanding crew comprised of those who served aboard it in the prime reality during 2267: Kirk, Spock, Scott, Chekov, Leonard McCoy, Hikaru Sulu and Nyota Uhura. Spock maintained a relationship with Uhura and reconciled with his father, Sarek, following the deaths of his Human mother and most of his own people. Ambassador Spock intended to found the New Vulcan colony for the 10,000 Vulcan survivors to live and thrive there. (Star Trek)
A fearful Federation
In 2258, Admiral Alexander Marcus became concerned about tensions with the Klingon Empire, and went in search for potential resources. He found the SS Botany Bay, recovering its seventy-three occupants. He awoke Khan Noonien Singh from cryogenic stasis, and blackmailed him into designing ships and weapons to prepare for the coming war.
However, a year later, Khan turned on Starfleet, coercing a member of Section 31 into bombing their base in London. During an emergency meeting in San Francisco, Khan attacked before escaping to an uninhabited area of Qo'noS using the confiscated transwarp beaming formula. During the attack, Admiral Pike was killed, and a vengeful Kirk was given permission to find and terminate Khan. The Enterprise was outfitted with seventy-two missiles, with the intent of using these to execute him from orbit.
When the Enterprise arrived at Qo'nos, Kirk opted to apprehend Khan, rather than firing a torpedo on his location. When they found him, Khan killed the Klingon patrol that had intercepted Kirk's away team and submitted to his authority. Aboard the Enterprise, Khan explained his involvement with Marcus, after McCoy and Carol Marcus discovered his crew were contained inside the missiles supplied by the admiral.
Marcus soon arrived on the USS Vengeance, and opted to destroy the Enterprise to cover up the conspiracy. Fortunately, Scott had stowed away on the Vengeance and deactivated its weaponry, giving Kirk and Khan time to space-dive and commandeer the ship. Khan then betrayed Kirk, killing Admiral Marcus, and threatened to resume bombarding the Enterprise unless his people were beamed aboard. The missiles were beamed aboard, but Khan reneged on the deal, forcing Spock to detonate the missiles; however, McCoy had removed the cryo chambers from the missiles before they could be used against the Vengeance.
Both ships were crippled and began descending due to Earth's gravitational pull. Kirk reactivated the Enterprise's warp core before it crashed, but at the cost of fatally poisoning him. When the Vengeance crashed in San Francisco, Spock beamed down to execute Khan in retribution, but McCoy realized Khan's blood could be used to revive Kirk, so Uhura beamed over to stun Khan repeatedly so that Spock could simply knock him out. McCoy then performed a blood transfusion, saving Kirk's life. Khan was placed back in stasis with the rest of his people.
Almost a year later, Captain Kirk presided over a memorial for the lives lost because of Khan and Marcus. The refitted Enterprise was rechristened and sent on the first, unprecedented five-year mission with the aim of promoting a less militaristic direction for Starfleet. (Star Trek Into Darkness)
The alternate reality runs parallel to the prime reality as a new quantum reality. The prime reality is where many of the events seen in the Star Trek universe have occurred and, according to Star Trek writers Alex Kurtzman and Roberto Orci, their film allows the prime reality to continue. 
This intent is also evident in the script of Star Trek.  While not completely audible in the film, before being teased by his classmates, young Spock is asked by the computer in the learning center on Vulcan: "What is the central assumption of Quantum Cosmology?" To which Spock replies: "Everything that can happen does happen in equal and parallel universes."
As the alternate reality is merely divergent rather than a completely new universe, this means backstory elements pertaining to anything before 2233 hold true for both timelines. Director J.J. Abrams said here that "It's actually nice when you're given a box.... when you're given parameters that you have to honor because it gives you limits and then you know that within those boundaries you can be creatively risky."
On the Star Trek audio commentary, the writers stated some events in the new timeline were meant to give insight as to what happened in the prime reality, such as how Kirk and Spock met following the Kobayashi Maru scandal. Orci opined that identical events would happen in both timelines because the "rules of quantum mechanics tell us that the universes that exist, they exist because they are the most probable universe, [...] the things that happened in the original series didn't just happen because they happened, they happened because it's actually what's most probably going to happen." 
However, when asked if this meant Spock and Uhura might break up as they were not in a relationship in the prime reality, Orci responded, "Their relationship is slightly predestined. On the other hand, our whole point was to give all of our characters free will again. They truly have free will. The universe is not written. The future is not written. And it’s not clear what’s going to happen. It’s going to up to what the characters do. Be it us as the next writers or someone else who has a better idea, may these characters fulfill their destinies according to their own devices and their own free will." 
Star Trek screenwriter Roberto Orci in a post on Ain't It Cool News  as well as in an interview with Star Trek Magazine  and J.J. Abrams in an interview with MTV conducted between the two aforementioned statements from Orci  establish a reason why technology in the alternate reality appears to be more advanced than it is during the same period in the prime reality. Scans and telemetry of the 24th century Narada, taken by the Kelvin, were brought back to Starfleet by the survivors on the Kelvin's shuttles. Therefore Starfleet's development and construction plans were slightly altered, making everything potentially more advanced, slightly ahead of schedule.
Name of the timeline
Startrek.com calls the new timeline in their online database interchangeably as the Nero-created "alternate dimension" , "alternate reality" , "alternate timeline"  and "alternate universe",  and differentiates the new versions of the characters introduced in Star Trek only with the production year (2009) of their appearance.
In an interview, actress Alice Eve called it the "split universe".  Simon Pegg joked that "I had this idea. I think the we might all be the mirror universe crew." Perhaps in the third movie, we'll see that "something's going to go to shit, we're all going to turn bad, Spock's going to grow a beard, and we're going to meet ourselves. That could happen." 
IDW Publishing calls the alternate reality the "new timeline"  or the "altered" timeline/continuity.  The board game Star Trek: Expeditions calls it the "new universe". Author of Star Trek novels Christopher L. Bennett, has referred to it as the "Abramsverse",  and noted the slang term prefix "Nu", which gave him "the impression that it was the label favored by people who wanted to dismiss the Abrams continuity or make it sound ridiculous." 
The Star Trek screenplay contains a musing from Spock Prime, having been told by Kirk that Chekov, Sulu, and Uhura were all serving in the Enterprise (by this point he takes it for granted that McCoy is as well), while Scotty is on the same planet as the two of them, that Kirk's implausible meetings with the people who would become his crew in the prime reality may be the result of the timeline trying to repair itself from Nero's damage. The film's novelization by Alan Dean Foster preserves this exchange.
Writers Mike Johnson and Tim Jones wrote Star Trek: Countdown, a comic book prequel to Star Trek expanding on the events in the prime reality leading to the Narada's arrival back in time. Johnson and Jones subsequently collaborated on Star Trek: Nero, based on the deleted scenes regarding Nero's imprisonment, and Star Trek: The Official Motion Picture Adaptation. Johnson writes the ongoing Star Trek comic, launched in September 2011, which explores how some classic stories unfold in the new timeline.
Four novels set after Star Trek were set to be published in 2010: Refugees, Seek a Newer World, More Beautiful Than Death and The Hazard of Concealing. The novels were announced as being on hold as of 14 January 2010. On his blog, Refugees author Alan Dean Foster speculated the hold was due to the plots of the novels possibly conflicting with the next film. Two years later, Christopher L. Bennett posted "All I can say is that most of the speculations I hear about why the books were pulled are wrong. It wasn't about conflicts with the second movie."  A series of young adult books, Star Trek: Starfleet Academy, following the characters during their time at the Academy, began in November 2010.
In The Needs of the Many, a novel based upon the Star Trek Online series, when Dulmur, one of the Department of Temporal Investigations agents from DS9: "Trials and Tribble-ations", is institutionalized and suffering from temporal psychosis, he has memories of not only his timeline, but of several others as well - including one where Vulcan was destroyed a century prior. Another novel Watching the Clock goes into detail about how and why some forms of time travel create parallel alternate realities and others lead to the overwriting of the same timeline. Bennett, the author, stated this was an in-universe explanation for the co-existence of the prime and alternate realities.