That would be known as a blown capacitor. It is most likely in the power supply, but could also be on the motherboard. If you look at the motherboard and there is one capacitor that does not look like the others, that could be it. This is just an example. http://www.pcstats.com/articleimages/200302/capblown_6....
Whether PSU or motherboard, you'll need a new one. I know how to fix such things, but few people do it for a living.
Unfortunately that machine seems to be rather old. According to my research it is an old Pentium processor and has DDR memory. If that is indeed the case, then it will be next to impossible to find a replacement motherboard.
All the spec says for the power supply is 250 watt ATX. You can still get a replacement for that, but I would not spend a whole lot of money for it. Certainly no more than $20. I do not normally advocate buying cheap power supplies, but it makes no sense to buy a more expensive one for this system.
My advice, if the budget allows, is to give this system to a recycler and buy a new low cost one.