(covers information from several alternate timelines)
The Jellyfish was a late 24th century starship commissioned by the Vulcan Science Academy for the transport of red matter. It was described as being "[their] fastest ship", and featured a rotating tail section.
After the ship's commissioning in 2387, Ambassador Spock piloted the Jellyfish during his attempt to save the Milky Way Galaxy from destruction by the supernova of the Romulan sun before it destroyed Romulus. Spock used the Jellyfish to inject red matter into the supernova, resulting in the creation of a black hole that absorbed the nova. While the plan succeeded, Spock was unfortunately too late to stop the supernova from destroying Romulus.
As Spock was attempting to depart, he was intercepted by the Romulan mining vessel Narada, and both ships were pulled into the black hole. The Narada emerged from the tunnel through space-time in 2233 and was the catalyst for the alternate reality, whereas the Jellyfish exited the black hole twenty-five years later. Upon capture by Nero, the ship was stored inside the Narada. Nero later ordered the use of the red matter aboard the seized ship to destroy the planet Vulcan.
The ship was destroyed when the younger Commander Spock of the alternate reality was able to retrieve the ship and piloted it on a collision course with the Narada, igniting the red matter inside. Spock survived the collision and was beamed aboard the USS Enterprise at the last moment. (Star Trek)
The Jellyfish was outfitted with a chamber designed for the safe storage and transport of red matter, as well as a means of extracting the matter for the purposes of its mission. In addition, the ship featured four forward-mounted torpedo launchers and a warp drive.
The ship was also equipped with the means to recognize its pilot, Ambassador Spock, via facial recognition and voiceprint analysis. As a result, it welcomed aboard Commander Spock and allowed him to operate its controls. It was at this point that he realized who must have helped James T. Kirk return to the USS Enterprise after being marooned on Delta Vega; asked if he would be able to fly the craft, Spock told Kirk that he likely already had. (Star Trek)
The ship was not given a name in the film, but it was named as the Jellyfish in the script, in which it was described as "a unique-looking spacecraft."  Concept artist Ryan Church envisaged the ship as having a transparent exterior, based on the actual translucent appearance of a jellyfish. Comic book artist Bryan Hitch contributed to the final shape of the ship, taking inspiration from a gyroscope. Church had another idea, of the ship being a cube of pure energy to emphasize its roots in Vulcan science, but J.J. Abrams felt the ship had to have "as much character as Leonard Nimoy had in his face".
Visual effects supervisor Roger Guyett stated that production designer Scott Chambliss envisaged the ship's exterior surface as "sophisticated technology married with organic things". "It might even be a technology Vulcans' 'grow', like a plant of high tensile steel." Guyett also said the ship's warp signature was intended to evoke clean "green" energy, in contrast to the "burned dirty fuel" aesthetic of the Narada. (Star Trek - The Art of the Film) The triangular pilot's chair and circular view screen form the Vulcan IDIC symbol when viewed from behind the main cockpit area.
According to information on the Blu-ray version, the ship is 178 feet (54.25 meters) long. The white bolts fired by the Jellyfish were identified in the script of Star Trek as torpedoes.  A dossier on the Star Trek movie website stated that the Jellyfish was a single-occupancy vessel with the maximum warp speed of warp factor 8.  This speed figure would indicate that the line about the Jellyfish being "our fastest ship" was not meant to indicate that the Jellyfish was the fastest Starfleet ship in 2387, but rather the fastest ship operated by the Vulcan Science Academy. The USS Voyager for example had a maximum warp speed of warp factor 9.975, sixteen years before the launch of the Jellyfish.
Interestingly, the computer aboard the vessel responded to Spock's inquiry in regards to its manufacturing origin using the stardate system of the alternate reality rather than that of its origin (the "prime" reality). This may indicate an Earth/Human-centric shift in the prime universe post-Nemesis.
Information originally posted on the Star Trek Online main website indicated that the Jellyfish was equipped with trans-metaphasic shielding. The experimental spacecraft was launched from the Daystrom Institute and Starfleet was working with La Forge and the institute to determine if some of the ship's systems could be adapted for Federation use.  Standard metaphasic shielding has been seen or mentioned in TNG: "Suspicions", "Descent, Part II", and in VOY: "Resistance". It is due to become a playable ship in December 2019 according to the Star Trek Online website.