The Umbilicus was a 370-mile-long cable that kept the Satellite of Love tethered to Earth during Season 6 and Season 7. The Umbillicus could be used to send objects from Deep 13 to the SOL and back, and seemed somewhat important for keeping the SOL in orbit (as well as keeping some of its functions on in a way). A similar device is used in Season 11.
Making its debut in the Season 6 premier Girls Town (as it emerged from the SOL to Deep 13, thus connecting the two), it was used relatively often in the show to transport things between the SOL and Deep 13 (which is signaled by a blue light riding across the tube when seen in action).
It was originally directly connected to Gypsy (thus making it an extension of her body technically at the time) with the objects being received and placed through her mouth, this seemed to be changed around the time of Colossus and the Headhunters and Bloodlust to an oven-like hatch on the bridge of the Satellite of Love (curiously where the airlock was), thus removing Gypsy's link from it.
At another point (Kitten with a Whip), Crow T. Robot was sent down to Deep 13 via the Umbilicus in an attempt to bring the SOL down, but was scared back up the SOL by Dr. Forrester, and Frank sets a giant mousetrap in front of the Umbilicus' Deep 13 entrance to prevent such escape attempts from happening again (although Crow went back to Deep 13 seemingly via the Umbilicus for a Thanksgiving dinner in the Turkey Day version of Night of the Blood Beast).
Sometimes the Umbilicus can set up for general stories, as at one point (Last of the Wild Horses), a matter transference device was transported up to the SOL during an ion storm, thus resulting in Tom Servo and Gypsy being displaced in an Mirror Universe (while their Mirror counterparts were displaced on the normal universe's SOL), it was not until the Mirror Dr. Forrester sends a matter transference device down his SOL's Umbilicus that things got back to normal.
The Umbilicus was disconnected when Dr. Forrester had to leave Deep 13 due to budget cuts at the Season 7 finale Laserblast. This somehow caused the SOL to slightly lose power and also cause its orbit to begin to decay, that was until Gypsy and Tom fixed thrusters onto the SOL which eventually saved all of occupant's lives (though at the cost of flying to deep space and eventually landing at the edge of the universe).
Since the SOL returned to an ape-ridden earth of the year 2525 in Revenge of the Creature (and it's further adventures from there), there was no attempt to create a new Umbilicus, plus the role of sending things up and down the SOL (as well as maintaining some of its functions) was given to Brain Guy (who uses his psychic abilities to do so, thus bringing flexibility of the size of the objects too).A similar device known simply as the Tube was made to connect the new SOL to Moon 13. Kinga Forrester used it to send Jonah Heston to the SOL when she captured him.