This is one of the most common hardware problems contributing to computer failure! It's simple to diagnose....your computer doesn't work properly. (Was that helpful?)
Actually, it is a bit more complicated than that. Your power supply basically, well, supplies power! Sometimes it gets old or develops a fault, but still gives out some voltage....except that your computer may not start or only "see" half the memory you just installed, because it doesn't deliver enough power to run everything properly. It's a bit like trying to run an aeroplane engine on ordinary 91 octane fuel.
Sometimes, your supply fails because you have added some extra components (memory, bigger graphics card, etc) and it can't deliver enough power to run it all. It's vital you choose one that delivers a greater Total Combined Output (TCO) than your old one. (Google the term) You should be able to find generic power supplies that don't cost the earth but deliver what they promise on the side label....not all do.
The best solution is to get a new power supply and install it. This is pretty easy, as connections are very straightforward. The main problem is getting a new supply that is up to the task!