The Supercomputer is a quantum supercomputer designed to hold Lyoko, XANA, and countless actual individuals in its data at the same time. It was built by Franz Hopper for the purpose of destroying Carthage. During construction of the supercomputer, Franz Hopper discovered the time reversion feature and used it 2,564 times to complete work on the computer. Each time reversion doubles the processing power of the supercomputer, and by extension XANA. The supercomputer is powered by a uranium rod. The actual supercomputer is at the lowest level of the Factory where it spookily rises from a hatch with a kind of liquid, most likely liquid nitrogen used for cooling, in the center of the room. The supercomputer also has a monitoring port a few stories above, but only Jeremie and Aelita can fully understand its potential.
Back in Season 2, XANA and Aelita were both dependent to the supercomputer's survival. Should it have failed, or ran out of energy, XANA would have been destroyed along with Aelita. (As seen in Common Interest) To that regard, the reason Team Lyoko did not shut simply shut down the supercomputer to cause XANA to deactivate because of Aelita's dependency to the computer.
The idea of the supercomputer doubling in power with every Return to the Past is probably erroneous - if the supercomputer added a qubit with every return trip, then after Franz was done its data-processing power would be laughable - most likely it takes multiple RTTPs to generate another qubit and increase the processing power. If the processing power truly did double every time, then the number would exceed ten googolplex.
The subject of the fuel rod is also of debate - Uranium's half-life means the battery would not expire for hundreds of years, also Duncan and Jeremie would have been severly poisoned from being around both enriched Uranium and the depleted fuel rod - therefore it seems lead-210, a uranium isotope with a twenty-two year half-life, is the used fuel - it only gives off particle radiation, which cannot penetrate the perspex sleath of the fuel rods, and which could be converted to energy by the same process we see in the scanners when a body is destroyed and rebuilt during virtualization and materialization, respectively.
In the end they did shut down the supercomputer in Echoes. Also, in this episode, it's hatch is seen with the XANA symbol on it.