This page contains information regarding Star Trek: Lower Decks, and thus may contain spoilers.
The USS Pegasus (NCC-53847) was a Federation Oberth-class starship operated by Starfleet. A prototype vessel, the Pegasus served as a testbed vehicle for various technologies, many of which would later be implemented on the Galaxy-class and other designs.
In 2358, the Pegasus was commanded by Captain Erik Pressman, and included Ensign William T. Riker, on his first assignment out of the Academy. In that year, the Pegasus was assigned to conduct a secret test of a phasing cloaking device, in direct violation of the Treaty of Algeron. During the test, an explosion occurred in main engineering, resulting in heavy casualties. Several members of the crew, including the first officer and chief engineer, mutinied in an attempt to shut down the illegal test. Pressman, Riker, and seven other crewmembers fled the ship in an escape pod. Not long after departing, the plasma relays were ignited with the ship still cloaked. The escaped crew, however, believed that the ship had exploded as a result of a warp core breach. In fact, the Pegasus drifted in phased-cloak form in the Devolin system until it failed inside asteroid Gamma-601, half of it materializing in solid rock. The rest of the crew died during these events, though the cloaking device itself survived.
The surviving crew was recovered and returned to Starfleet. Subsequent investigations found no wreckage, and the crew, led by Pressman, stated that the accident in engineering had been responsible for the destruction of the Pegasus, and made no mention of the cloak. The Judge Advocate General's inquiry into the incident concluded that the surviving officers had not been entirely truthful in their recounting of the events, and suggested a further investigation. The follow-up investigation never took place, however, and the inquiry report was quietly buried by Starfleet Intelligence.
Twelve years later, in 2370, the Romulan warbird Terix discovered a piece of wreckage from the Pegasus in the Devolin system. Pressman, then a rear admiral, was given authority to dispatch the USS Enterprise-D to recover the vessel first, aiming to reach the cloaking device before the Romulans discovered the Federation's duplicity. The Enterprise entered the asteroid and found the ship. The illegal events that happened aboard were made public, and Pressman, among others, were court-martialed. The ship itself was not recovered. (TNG: "The Pegasus"; ENT: "These Are the Voyages...")
A decade later, while serving as captain of the USS Titan, Riker displayed a golden model of the ship on his ready room desk alongside a American Civil War-era military cap from the 102nd New York Infantry. (LD: "Kayshon, His Eyes Open")
|USS Biko • USS Bonestell • USS Cochrane • USS Copernicus • USS Grissom • USS Oberth • USS Pegasus • SS Tsiolkovsky • SS Vico • USS Yosemite • Unnamed|
In the episode, La Forge asks "What would the Romulans want with pieces of a twelve year old starship?" suggesting that the Pegasus was also launched in the same year it was lost. However, Pressman claimed to have known most of the Pegasus crew "a lot longer" than Riker had, implying they had served under his command for much longer than that.
The original model of the USS Pegasus (USS Grissom) is now part of ScienceFictionArchives.com collections. 
According to the script of the episode "Too Short a Season", the USS Pegasus was one of the ships that had been commanded by Mark Jameson. In the script, just before Jameson beamed aboard, Jean-Luc Picard mentioned that he had heard legends of Jameson's command of the old Pegasus in the Klingon conflicts. The scene mentioning the Pegasus was cut from the final episode.
Rick Sternbach designed a new ship type in October 1993 for the Pegasus based on parts from the USS Enterprise-C. Budget limitations forced the re-use of the Grissom studio model from Star Trek III: The Search for Spock. (Star Trek: The Next Generation Companion (2nd ed., p. 278)) Sternbach himself has noted in this regard, "As far as the Pegasus was concerned, yes, there were a few sketches of a ship done, but to make it easy on the VFX folks it was supposed to be a kitbash of sorts from either a Nebula or an Ambassador (I don't have the sketches handy, so I can't recall exactly). For whatever reason, probably time/money, the Oberth miniature or a copy served as the Pegasus."  Though not apparent on his sketch, it appears that this design would have ended up with four nacelles, as an engineering display visible in the final episode shows such a configuration.
Rick Sternbach's original sketch (titled "The Pegasus" Ship Concept Semi-ORIGINAL Sketch) was auctioned off on eBay on December 1, 2009. The relevant part of its description was reproduced below:
This was an 11" × 17" semi-original pen and ink and pasteup drawing for a concept for making the U.S.S. Pegasus out of Enterprise-C physical model parts, instead of re-using the model of the U.S.S. Oberth science ship. Drawing is signed in the lower right. 
Based on his original design, Sternbach had also produced a detailed master systems display graphic in ink of the design for use in the episode, which ended up unused. Further commenting Sternbach added, "I think the ink drawing assumed Ambassador parts, from what I recall of the general proportions. I don't recall if I did a new systems cutaway for the Oberth-Pegasus."