computer doesn't startup

I have a Gateway E-600 computer with pentium 4 at 1.6GHz. I had this computer for 5 years. A couple of weeks ago it would not startup. What happened is that when I turned on the monitor and the desktop, the monitor did not go to the startup screen and just went into power saving mode. The screen test said it worked correctly and I checked the connections in the back, but there were no loose connections. The power button on the desktop was green. Besides that it did not do anything.

Usually when I start up the computer I would hear the dvd, and floppy drives come on and see the little lights in the front blink. Also I could hear the harddrives come on as well.
When the problem occurred i did not hear anything activate in the desktop.

I thought I solved this problem by disconnecting the cord from the power supply and plugged it back in. The same problem happened again today, but the solution I used before did not seem to work. I had to push the power button a couple of times to get it to work correctly again.

Can someone please tell me what the problem is with the computer and how to fix it?Thanks for any suggestions.