Borg spheres were used by the Borg as scout ships or long-range tactical vessels. Borg sphere were also embedded into some Borg cubes, and were used as auxiliary craft. (Star Trek: First Contact; VOY: "Dark Frontier", "Drone") Borg spheres had a crew complement of 11,000 Borg drones. (VOY: "Unimatrix Zero, Part II") The spheres were approximately 600 meters in diameter, and had an interior bay large enough to hold an Intrepid-class starship. (ENT: "Regeneration"; VOY: "Endgame") Borg spheres had transwarp capability and ablative hull armor. They were also equipped with a tractor beam. (VOY: "Drone")
Provided its deflector shields were down, a Borg sphere could be destroyed by a Sovereign-class starship with relative ease using quantum torpedoes. (Star Trek: First Contact) In 2378, Starfleet attempted to destroy a sphere with a fleet of starships. The sphere was holding the USS Voyager, which was able to destroy the sphere from the inside using transphasic torpedoes. (VOY: "Endgame")
After the Battle of Sector 001, a Borg sphere was observed to have time travel capabilities, by generating a temporal vortex through the controlled emission of chronometric particles. The Borg sphere traveled back to the year 2063, in order to prevent the historical event on Earth, known as First Contact. (Star Trek: First Contact) In 2153, a team of scientists found the remains of the Borg sphere from 2063 in the Arctic Circle. (ENT: "Regeneration")
In 2375, the crew of the USS Voyager successfully retrieved a Borg transwarp coil from a damaged Sphere, as part of Operation Fort Knox. At the time, the sphere was only able to travel at warp 2 because it was regenerating from the damage. (VOY: "Dark Frontier")
A sphere had no living quarters, or a discernible engineering section for the propulsion systems. There however was a primary shield generator inside the shield matrix. The several transwarp coils of a Sphere were contained inside specialized rooms called transwarp chambers that featured heavy shielding. (VOY: "Dark Frontier")
Ships of the Class
The physical model created by ILM was actually painted dark indigo, despite its dark gray appearance on screen. That this model's appearance is similar to the Death Star from the Star Wars films is not just coincidence. ILM did all of the effects and modeling for Star Wars. However, designers of the Borg sphere did make a conscious effort to make the sphere different from the Death Star in order to avoid too many comparisons. The schematic of a sphere shown in VOY: "Dark Frontier" depicts the most striking resemblance.
The official size comparison chart from Star Trek: First Contact defined the size of the vessel as being 1500 feet (457 m) in diameter, however, according to the size of the debris field, as referenced in "Regeneration", the radius was calculated at "about 600 meters in diameter."