Hi Eric,
It sounds like the computer is receiving power, but not able to POST when you press the power button. This could be any number of problems, really; maybe bad RAM, a bad CPU, or possibly a bad motherboard.
There's no simple fix to this. You can try moving the memory modules, if possible, or replace them altogether. If no change, you will need to have a technician troubleshoot the hardware to determine what's causing the system to auto shutdown.