My computer reboots randomly and comes up saying that my 'computer has just recovered from a serious error'. What does this usually mean, if it could be narrowed down to a couple of things?

If it does it at random times and just completely shuts off instead of shutting down each time, then it is most likely the power supply. They are about $80-100 for a decent one and are very easy to replace. Just about anyone can do it.

It is probably giving you this error because it was not shut down properly.

All the above posts are pretty much right on the money, A few other possibilities do come to mind though.
A. you could have a processor fan not running causing the processor to overheat. This would cause a restart.
b. A stick of ram going bad can cause this as well.
c. Anything from a irratic cd drive, to a modem, ethernet card, being not fully seated to the motherboard, faulty motherboard, bad power supply, etc.
d. faulty hardware driver.
e. virus
f. A failing harddrive.
the recovered from critical error is because the unit simply restarted without a proper shutdown.
You most likly can recreate this by cycling the power (hard kill) while the unit is powered up. (Not advisable).
Bottom line is get it checked out as this can lead to loss of data as well as operating system damage.

Unlike application errors, these typically happen when something in the system kernel falls over. Usually this is something important like a device driver and will either be the result of poor coding or a hardware fault. It could also be some malicious software that is hiding somewhere important as a driver or rootkit.

Typically there is more information in the error message that can lead you to what caused it.

It may be the powersupply or damage to the motherboard. A number of older (2-3 year old) Dell PCs have faulty capacitors that have to be replaced around now it seems.

Id be a bit worried as the pc shouldnt just reboot for no reason without you allowing it too,maybe you have a virus!