The Mothership was positioned above the Earth, orbiting much like a satellite. It served as the base of operations for Pox and Crypto's mission on Earth. In between missions, Cryptosporidium would return to the Mothership to rest and obtain updates in between missions.
The Mothership looked like a mushroom with a fatter base, floating above earth with a purple beam of light coming from the bottom to guide Crypto around the surface of Earth.
The Furon Mothership was a space station and a ship. It could travel interstellar distances in short amounts of time. The station had extensive arrays of sensor devices, for tracking Humans and keeping tabs on Crypto, even from star systems away. It also has a stock of weapons and ships to spare for Crypto if he is killed in battle.
The ship housed Gastroenteritis, Orthopox, their living quarters, an advanced science laboratory for researching upgrades, and the war room where Crypto is directed on missions. It didn't seem to move around much except from planet Furon to planet Earth. However, its outer mechanisms rotate while in orbit. Since Furons are warlike, it is most likely armed with powerful weapons of mass destruction that were never revealed.
It was heavily implied in the extras of the first game that the Mothership is filled with many Crypto clones. With Furon cloning technology, they could create an endless number of Crypto clones to use as soldiers. This would give them an unstoppable army that they could use in their planetary conquests if necessary.
Crypto and Pox's Mothership was built during the early Martian War, and belonged to them for a long time before they were given their harvesting mission. They most likely use it to travel through space to conquer worlds in the name of the Furon Empire.
In Destroy All Humans!, it acted as their base for the mission to harvest the Furon DNA of Humans. It wasn't used for much else other than a rest stop in between missions, research into upgrades, and detecting objects and locations of interest.
In Destroy All Humans! 2, it was destroyed by a KGB nuclear missile, which also killed Orthopox and made him into a hologram. The destruction of the Mothership was a major inconvenience to the Furons, without it, their armory was understocked, and their grip on the planet Earth was at serious risk. They tried requesting a new one from the homeworld, but the admiral got angry for their failure to preserve the DNA and he would've authorized another fleet commander to replace Pox. Of course, Pox's pride wouldn't let someone else take all the credit for the honor of being the savior of the Furon race, and is resolved to continue the mission without a replacement Mothership.