This page contains information regarding Star Trek: Discovery, and thus may contain spoilers.
In 3188, Booker's ship fled Hima while under fire from Cosmo Traitt's ship, as Booker had stolen a trance worm he dubbed "Molly" from Traitt. Suddenly, Michael Burnham emerged from a wormhole in the Red Angel suit and collided with Booker's ship, causing both to plummet to Hima. After Burnham helped Booker obtain more dilithium from the Mercantile, he transported Molly to Sanctuary Four and Burnham to Sahil's relay station. (DIS: "That Hope Is You, Part 1")
A year later, Burnham was on Booker's ship when she received a signal from USS Discovery. The ship employed a tractor beam to pull Discovery free from parasitic ice on the planet where it had crashed. (DIS: "Far From Home")
The Discovery subsequently jumped via spore drive to Earth with Booker's ship in its shuttlebay. To avoid unwanted attention, the Discovery crew hid their store of dilithium on Booker's ship under cloak. This would allow Burnham and Booker to trick and capture Wen. (DIS: "People of Earth")
Later that year, Booker landed on Hunhau in search of a black box from the USS Giacconi, and was captured by Tolor. The ship autonomously set course back to Discovery with his cat, Grudge, and arrived at Federation Headquarters three weeks later. Burnham and Philippa Georgiou returned to Hunhau aboard the ship to rescue him, along with the other slaves present, and destroyed the facility. (DIS: "Scavengers")
When Osyraa's flagship, the Viridian, began bombarding Kwejian later that year, Lieutenant Detmer and Ryn engaged the heavy cruiser in Booker's ship so that the Federation would retain plausible deniability. Booker's ship took eight photon torpedo hits, but was able to drive the Viridian off before returning to Discovery for repairs. (DIS: "The Sanctuary")
Following Osyraa's capture of the USS Discovery, the ship reconfigured into a dart-like configuration and was used to board the Discovery via a crash landing in the shuttlebay, taking severe damage in the process. (DIS: "There Is A Tide...")
The ship had the ability to "morph", meaning that its sections could disassemble and assume different configurations in mid-flight. It was equipped with directed energy weapons, a powerful tractor beam, and a cloaking device. The main faster-than-light propulsion system was warp drive, though the ship could also utilize quantum slipstream (given benamite) or tachyon solar sails. It had a dilithium recrystallizer. Subspace communications systems included a transmitter, receiver and amplifier. (DIS: "That Hope Is You, Part 1", "Scavengers", "That Hope Is You, Part 2")
The ship had a single deck. The main cabin area featured a horseshoe-shaped pilot station with programmable matter controls and holographic displays, facing an L-shaped viewscreen port. Entities on the screen were projected as raised-pinscreen holograms. Certain components of the ship's systems descended from the ceiling in protective cases, rising and lowering while the ship was in flight.
Just off from the pilot cabin was a wood-paneled sleeping area with storage shelves shaped into the walls. Another nearby alcove offered additional storage.
Outside the main cabin was a paneled hallway, featuring hexagonal doorways and window openings between compartments with vertically-oriented curved transparent bars.
While not named on screen, StarTrek.com has published fan-contributed episode recaps that have used the name Nautilus.       In the episode "Anomaly", the ship was identified with this label "Vessel - C. Booker" in a graphic, ending any speculation on the ship's name as it had no name in the canon.