i assume we're talking about a desktop, not a laptop?
Right - Basically, there is a little battery which is on the motherboard (CMOS battery) which assists in ensuring the computer's BIOS (time, boot settings etc) stay the same when you turn your computer off.
it looks like the battery is dead and is no longer supplying power when the computer is turned off. I would recommend you replace the CMOS battery.
This can be done yourself.
Turn the computer off but leave the power cable in to earth the computer.
remove the side panel
Locate the CMOS battery, which is a 3V circular disc battery (like you'd get for a watch).
Replace this battery with another 3V battery.
These batteries are very cheap.