Why would a PC reset itself or shut down on its own?

A friend and I are trying to solve a problem with her PC. She just got it, but it’s old, and running stuff from like, 2002. It has XP, it’s a P4 and we don’t know much else. Lol. The person who had it before has a bunch of games installed on it, and doing anything with them makes the computer reset itself. When trying to download something, it shut off, and wouldn’t even turn on again…the power light was on, but nothing. We had to unplug it, and plug it in again.
I know it can be many things, but from these details, does anyone have any idea what’s wrong with this machine? We got the latest version of FF on it, and can’t download AVG because it shuts off…or at least I’m thinking that’s it, but I don’t know…seems really random. If I had to guess, I’d say there’s something wrong with how information is sent, processed or received…but it really seems to just happen when we mess with the games or trying to install things. (I’m guessing upgrading FireFox isn’t ’‘installing’’ anything, as opposed to the anti virus)
But I don’t know anything about PC’s and neither does my friend. We did our best to try and figure it out, but maybe some people here can help? Any ideas?