In the Teenage Mutant Ninja Turtles comic books and television series, the Technodrome is a mobile fortress and command centre for Krang, ruler of Dimension X. It is a white sphere with a tower on top resembling an eyeball, and with tank-like treads that propel it on the ground. Aside from its paint job and the eyeball and treads, it resembles a Death Star without its super-laser. It boasts impressive energy weapons of its own, however.
In the TV series and comic books, the fortress stood six hundred feet tall1 and posessed (among other Dimension X technological artifacts) a dimensional portal used to transfer items to and from Earth and Dimension X.
A toy playset is also available, but not nearly to the extreme physical dimensions of the TV series version, standing roughly three action figures (fifteen inches) tall.2
The pilot series of TMNT featured the Technodrome on Earth, when The Shredder and Krang plotted to bring an invasion army from Dimension X. A timely intervention by Donatello caused the fortress' portal to work in reverse, pulling itself and its occupants to Dimension X, where it stayed for the entire first season.
Some time during the second season, our nemesis Krang succeeded in returning the fortress back to Earth, where it had power problems that plagued it for the entire season (and an endless supply of plot devices to go with it. Michaelangelo (the character, not the E2 user!) sums it up thoroughly: "They've been after power for the Technodrome in every episode!").
Eventually, the bad guys restore the Technodrome to full power (One of Rocksteady and Bebop's few successes), and nearly destroy three miniaturized military bases before being miniaturized themselves by Krang's own Minimizer weapon.
Some time after that, after a plot to pull the entire planet Earth into Dimension X, our heroes find a way to reverse the process again, putting the Technodrome back where it belongs. Best as I can remember, it stayed there for the rest of the TV series.
1. From an episode where the fortress rampaged its way through Manhattan and faced the National Guard. http://www.thetechnodrome.com/archive/members2.boardhost.com/ks_suggest/msg/2.html
3. Based on the episode list available at http://www.thetechnodrome.com/eplist.shtml