When I plug the power cord into the wall socket, my computer turns itself on.

I mean I don't have to push the power button in to make the computer turn on.

When it DOES turn on, the processor fan spins only for a second or two. The power l.e.d as well as the motherboard light is lit but nothing else happens.
I can't hear the hard drive spin up. The DVD-rom drive does not blink.
The display (CRT/LCD) screen is blank. There is NO post screen (no data or words ever appear on the monitor).
If I hold the power button for few seconds only then the power l.e.d turns off. There are NO beeps when the computer is first turned on with the power switch.

It just sits there, doing nothing, blank .

Any changes that either happened or that you perhaps made to your computer around the time that the problem started; particularly any software or hardware that you installed or setup.

About your computer turning itself on when you just plugged-in to the power outlet, perhaps you have enabled the Turn on after power outage or something similar in your BIOS settings. Just disable it.

About your computer not booting up, make sure nothing is loose inside your CPU (RAM, video, audio cards, power cables, data cables... ) Just make sure it is unplugged from the power outlet when you do. After making sure that nothing IS loose and your computer still doesn't boot up, perhaps you should have your power supply or motherboard checked into.