If your computer turns off upon startup, here are a few troubleshooting tips to try.
If your computer is a mobile computer such as a laptop, netbook, or tablet, you will want to first check on the status of the battery to see if it is charged or drained. If the battery is drained, let it charge for at least 30 minutes before attempting use.
A dead motherboard tends to present itself as a failed startup. The Power button (if equipped) with flash once and then immediately turn off. Also, you will not hear the power supply unit (PSU) engage as well. To confirm the motherboard is dead, press the tab on the PSU while it is plugged in and see if it engages. If it does, you will need to swap the motherboard or invest in another computer that will accept the parts from your dead system and transfer over those useable parts.
Power Supply Unit
The PSU is always in use and tends the be the third item (behind the HDD and monitor) in your computer system that will fail. To confirm that it is functional, listen for sounds from the rear and breeze being pushed as the blades rotate.