(covers information from several alternate timelines)
Quantum reality, or quantum universe, was the term used to describe the alternate timelines in which all possible outcomes for any event take place. Each quantum reality was its own separate parallel universe characterized by a unique quantum signature which could not be altered. All matter resonated on a quantum level with this signature within the universes. Before 2370, the existence of different quantum realities was considered only a theory in quantum physics. Under normal circumstances quantum universes remain separated from one another by barriers. (TNG: "Parallels")
In 2370, Lieutenant Worf piloted the type 6 shuttlecraft Curie through a quantum fissure, which acted as a "keyhole" that intersected multiple quantum realities. The warp nacelles of the shuttle caused a small break in the barriers between realities and threw Worf into a state of quantum flux. The flux was intensified when Worf was near the subspace field pulse of Commander La Forge's VISOR. This had the effect of pushing Worf into new realities.
Eventually Lieutenant Commander Data was able to determine that, by piloting the same shuttle with which he had traversed the fissure in the first place and initiating a broad-spectrum warp field, Worf could seal the fissure and return to his own universe. This had the effect of transporting Worf back in time to a point before he encountered the fissure. However, as Data explained, due to the uncertainty principle, it was just as possible Worf would have ended up several days forward in time. (TNG: "Parallels")
Several different quantum realities were witnessed by Worf in 2370.
The writers of Star Trek, Alex Kurtzman and Roberto Orci, were asked about the implications of the new alternate reality that was introduced in the film in an interview. They explained the new reality runs parallel to the prime reality as a new quantum reality, as the concept was explained by Data in the episode "Parallels". 
This is also referenced in the script of Star Trek, 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."  In the final cut of the film, this is barely audible.
While not specifically confirmed, DS9: "Meridian" suggests that parallel universes on different dimensional planes are synonymous to quantum realities. In the episode a solar quantum level cascade causes a quantum fluctuation in the molecular structure of the planet Meridian and its inhabitants. The fluctuation shifts them to a different dimensional plane.
According to the 2008 novel Fearful Symmetry by Olivia Woods, the mirror universe is also a parallel quantum universe, as quantum signature scans used to match Worf with his USS Enterprise-D in "Parallels", were also able to differentiate natives of the mirror universe from those of the prime reality. The mirror universe has canonically been described as a parallel universe on a different dimensional plane.
Several other quantum realities are introduced in the Star Trek: Myriad Universes series of novels and comics. According to the story Places of Exile by Christopher L. Bennett in the 2008 novel of the series, Infinity's Prism, quantum realities are in fact synonymous with all types of alternate timelines.