(covers information from several alternate timelines)
The USS Franklin (NX-326) was a 22nd century Federation Franklin-type starship operated by Starfleet. It was the first Earth ship capable of reaching warp factor 4. Its armaments, typical of the era, included pulsed phase cannons and spatial torpedoes. Its defenses included polarized hull plating. Auxiliary craft included a complement of shuttlepods. (Star Trek Beyond)
Following the Earth-Romulan War and the formation of the United Federation of Planets, the Franklin received a new dedication plaque and was placed under the command of Captain Balthazar Edison, a former MACO veteran of the Xindi and Romulan wars. The ship vanished in 2164 while on a mission in the Gagarin Radiation Belt, an incident later taught to future Starfleet officers at Starfleet Academy. Speculation abounded about the ship's disappearance, with theories ranging from it surrendering to the Romulans to being captured by a "giant green hand".
However, the distance traveled suggested a wormhole displacement. The ship eventually crash-landed on the planet Altamid. As no sign of rescue came, Edison came to believe the Federation deliberately abandoned them. Edison discovered relics left by Altamid's native species, including numerous ships, a drone workforce, and life-sustaining technology capable of stealing the "life force" from its victims. Edison and his crew abandoned the wreck of the Franklin and turned against the Federation.
By 2263 of the alternate reality, the Franklin had been discovered, half-buried under the surface of Altamid, by Jaylah, who placed it under a holographic cloak to use as a residence. The Franklin had functioning power reserves, serviceable engines, and intact data logs despite a century of neglect. Before her encounter with the crew of the late starship USS Enterprise, Jaylah had begun repairs on the Franklin, learning English from its on-board computer and becoming familiar with the "classical" music of 20th century Earth stored in the ship's data banks.
Marooned on Altamid, Captain James T. Kirk, Commander Spock, Lieutenant Commander Montgomery Scott, and Ensign Pavel Chekov further repaired the Franklin, restoring its transporter capability, and, enhancing it to transport complex lifeforms rather than simply cargo, they rescued the Enterprise crew imprisoned by Krall. Learning of Krall's plan to use his swarm ships and the super weapon known as the Abronath to attack the Federation Starbase Yorktown, Kirk and company successfully piloted the Franklin out of Altamid's atmosphere by dropping the ship off a cliff in order to reach terminal velocity (as the ship had never been designed to land, they had to let it free fall in order to create enough forward momentum to achieve escape velocity).
Encountering Krall's forces at Yorktown, Kirk and company piloted the Franklin into the swarm. Using the ship's antiquated technology, Kirk's crew disrupted the individual vessels' communications with radio signals and created a chain reaction that destroyed most of the swarm. Pursuing Krall and two remaining ships, the Enterprise crew piloted the Franklin into Yorktown's internal docking structure, using the ship itself to destroy the attacking vessels before crashing into Yorktown Central Plaza near Yorktown Headquarters. In the aftermath, Kirk and Commodore Paris officially closed the missing-in-action report on the Franklin and its crew. (Star Trek Beyond)
The USS Franklin was named after director Justin Lin's father, Frank Lin, and its dedication plaque has a slightly wider gap between the k and l to further emphasize this tribute. The Franklin's registry number 326 was in honor of Leonard Nimoy's birthday. 
The use of the NX prefix and USS would seem to have been added after the formation of the Federation, as United Earth starships depicted in Star Trek: Enterprise did not commonly use USS and the NX prefix was only used for ships in the NX-class or part of the NX Project and Warp Five program.
The ship was designed by Sean Hargreaves, who described it as initially being "almost like a large shuttlecraft". Regarding its resemblance to Doug Drexler's NX-class Enterprise, Hargreaves stated that the Franklin was designed independent of that ship, initially featuring nacelles in the down position before the script required them to be flipped above the saucer section for the takeoff sequence. Hargreaves also explained that the pennants on the Franklin's engines depicting the Starfleet insignia were originally only placeholders that were never changed. According to the designer, the Franklin was intended to resemble a freighter and was 450 feet long. 
In 2016, collectible miniatures of the ship were produced by Snapco and QMx as home video and theater concession stand premiums. Three more miniatures are slated for release by Hallmark, QMx, and Eaglemoss Collections' Star Trek: The Official Starships Collection partwork magazine in 2017.
Regarding the vessel's origins, Dylan Highsmith said, "If you want the official explanation on the Franklin and its warp factor: it was a MACO ship (or a United Earth Starfleet ship that housed MACO personnel at times) that predates the NX-01. When the UFP Starfleet is formed, MACO was disbanded and the ship was reclassified as a Starfleet ship [with the "USS" identifier]. The ship is then 'lost' in the early 2160s. It was important to everyone that the ship, like Edison, predate the Federation; that thematically, the ship mirrored an earlier time in history and served as a bridge in design between then and the NX-01. Doug and Simon may have worked up something [on an official launch date], but if they did it never made it to script or screen. Either way it predates the NX-01, and was reclassified after the UFP is formed." 
The possible launch date of the Franklin as Earth's first warp 4 vessel can be narrowed down based on information from ENT: "First Flight", which established that the warp 3 barrier was first broken by the NX-Delta in 2145, and that the construction of Enterprise, Starfleet's first warp 5 ship, began in 2150. Therefore, the launch of the Franklin likely occurred within that range.