The unnamed Spectral Space Pirates' ship was the time vessel the Space Pirates used to travel through time and space.
When it collided with the Backlander in the Time Vortex, its engines were knocked out, which caused them to collide again later (as Foolscap pointed out). After the defeat of the Dark Pterosaur, its engines were again damaged, only being capable of making one last jump through time. It was used by Jonathan to bring Rex, the Ancients, the Alpha Gang, and all the dinosaurs aboard back to the future.
Like the Backlander, the Space Pirates' ship was outfitted with a cloaking function and, as seen in Fate of the Cosmos, was also capable of fixing any damage caused since its arrival in a given time period. It is capable of docking with other vessels using the tip of its nosecone.
Unlike the Backlander, however, the Space Pirates' ship had a working time-jumping ability throughout most of its appearances, getting the Space Pirates to the intended time period first, though their early start rarely proved fruitful. It was, unsurprisingly given their name, capable of traveling into space, often maintaining a low Earth orbit while the Space Pirates searched for the Cosmos Stones. It also possessed offensive weaponry, specifically what appears to be laser beams capable of zigzagging.
Somewhere onboard was the room where Max's house was kept. Where this was in relation to anything else on the ship was never explicitly shown, though there are shafts leading from it to one of the one or more large rooms of dinosaur cages. From here the Space Pirates would presumably take the dinosaurs they were to use in battle, though surprisingly none of the dinosaurs seen onboard were ever actually used by them.
Onboard was also apparently a docking bay or hangar for several smaller, also unnamed pod-shaped ships. These ships were capable of flying and landing using four forward-pointing spike-shaped wings with jets on the end. A main engine was located on the back of the craft for primary propulsion, likely leaving the wing jets mostly for pushing the ships backwards to take off. It has room for one pilot and one passenger.