First it would stop working because everything is designed to work in the commercial temperature range (0 to 70 degrees C). One would hope it works down to 0 degrees, although if the designers were not dillegent it may not make it down that far.
It would stop working because the various bits would fall out of specification, and each would do so differently. It's hard to say what would fail first. It would be one of:
- There would be so much timing skew between the signals that the various digital bits were receiving and what they required at that temperature that interfaces (i.e., between memory and processor) would stop to work, or logic would get scrambled.
- Oscillators would stop working because the gain necessary to maintain oscillation wouldn't be there.
- Voltage references would generate the wrong voltage (and not predictably, at least not up front).
- Transistors in the power supply circuits would fail to conduct properly.
There's a vanishingly slim chance that it'd still work below 0 degrees C. I'm starting to guess here, but there's yet more that can go wrong:
- Anything spinning, such as disks and fans, will seize up because of differential thermal contraction. They may be OK when they warm up again, but depending on how the disks stop working they may damage their surfaces.
- If the temperature goes down far enough, differential thermal contraction between various bits (i.e., the traces and the PCB, the leadframes in the IC's and their epoxy packaging, the silicon chips themselves and either the leadframes or the packaging) will cause things to crack. This damage would be permanent.
- If this is done in atmosphere, water vapor (followed by CO2, and then the various other constituents of air) would condense on the PC. Oxygen would condense out before nitrogen, which may cause interesting chemical reactions.
Basically, long before anything superconducts, the 'puter will have stopped working, and will very probably be permanently damaged.
Note that you can design electronics to work at cryogenic temperatures -- I used to work in IR imaging, where the imaging array was running at 77K, with CMOS circuitry. But that was circuitry that was specifically designed for the task, and which did not work well (or at all, sometimes) at higher temperatures.