Is the fan turning? (I'm assuming this is a desktop and not a laptop)

Power supplies losing their fans can be a pain - they'll work just until the unit gets too warm then shut down. Same with CPU fans. And they can be temperamental with power outages. (especially if you generally leave your computer on all the time)

Both of which you can get at Best Buy (or even Staples) and if you're at all mechanical you should be able to swap it out.

(if you lived nearby I'd offer to look at it for you- I used to do desktop computer hardware for a living)