The unnamed Spectral Space Pirates' ship was the time vessel the Space Pirates used to travel through time and space.
This vessel was used to take the Space Pirates to every destination across time searching for the Cosmos Stones. It was controlled (and presumably piloted) by Spectre. It was usually seen during transition scenes floating in orbit around Earth and rarely did anything active throughout most of the season.
When it collided with the Backlander in the Time Vortex while pursuing the fleeing traitor Seth, its engines were knocked out, which caused them to collide again later (which Foolscap pointed out the irony of). 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, abandoning the Backlander whose engines were completely broken.
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, whose ceiling seemed to provide a view of the Time Vortex during travel. 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 (with the exception of the Carcharodontosaurus seen in an unplaced flashback in Bad Deal).
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 (in a second seat at the back sometimes shown instead of the rear engine). Owing to its advanced design and construction, it is shown capable of crashing and only taking minimal damage, though there are limits, such as dinosaurs falling on them.
There is a modified setup of the pointed wings with Spectre's throne mounted on it that is used to hold the Cosmos Stone Detector in late episodes. Though destroyed by volcanic rocks, another one is later seen in use collecting the Cosmos Stones.