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's clones, 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.
Their Mothership was also mentioned by Pox to have a main invasion force that is air-based when explaining the danger that the Tesla Coils pose to it.
Crypto and Pox's Mothership was built during the early Martian War, and was theirs for a long time before they have gone on their harvesting mission. They most likely use it to travel through space to conquer worlds in the name of the Furon Empire.
During the Furon Invasion of Earth in 1959, it acted as their base for the mission to harvest the Furon DNA of Humans. It wasn't used for much else other than to rest in between missions, research into upgrades, and detecting objects and locations of interest.
In 1969, the Mothership was destroyed by a KGB nuclear missile in the beginning of Destroy All Humans! 2, 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.