450W generic power supply
Pentium 4 processor
2GB DDR2 RAM
Gigabyte G31M-ES2C mobo
Western DIgital Blue 500GB HDD
No GPU (mobo graphics which is about 64MB)
So, you've done already all that might be done by a user with a rather advanced knowledge and actually, from what I understand, it doesn't even show the BIOS startup
Looks like you've got a serious failure of the motherboard, probably on the power input level. Might not be of great help but check the condition of your 450W PSU; a problem with the PSU might have damaged the motherboard. Can't really give you more explanations about all this, because even if I have some experience with computers gone wrong, I don't have any computer electronics knowlegde.
Maybe some advanced electronic tests might diagnose your problem but I guess you should have the technical preparation and equipment to do it yourself.
By the way, your motherboard looks rather old so if it has failed, might not be worth to repair it, even if it were possible, which is frequently not the case.
My advice is:
Option 1: get yourself another motherboard compatible with P4 and DDR2. Before installing it, be sure to check your PSU.
Option 2: get a good computer compatible with your DDR2 memory. There still are lots of old P4 computers around with 1 Gb or 2Gb RAM and 100 or 200 Gb HDD
Option 3: Get a new computer. You can install your WD 500Gb as a second drive unit or as an external drive in a USB enclosure. As for the DDR2, the P4 and the 450W PSU (if it's good) you still might sell them on a garage sale for some money, though really are not worth much.