The Sony PS3 is more or less just like computer. The main operating system that is run on the PS3 is based off of a Unix kernel, which means it is based off of Linux. Like a computer, the PS3 has a motherboard that holds and connects its circuits together. If at some point any part of the motherboard breaks, the PS3 will lose most--if not all--of its functionality. Sony has designed the PS3 in such a way that the evidence of a broken motherboard is evident without needing to remove the PS3's case.
Plug one end of the PS3's power supply into a wall outlet and plug the other end into the back of the PS3. Push the main power switch in the back of the PS3 to send power to the PS3. This will not turn the PS3 on.
Turn on the Sony PlayStation 3. Wait for the PS3 to completely turn on. If the PS3's power light blinks green, then blinks yellow and continues to flash red, the PS3 motherboard is more than likely dead.
Hold the "Power" button on the front of the PS3 and the PS3 will enter its recovery menu. If the PS3 does not enter into the recovery menu, the PS3 motherboard is broken beyond repair. It will either need to be replaced, or Sony will give you a refurbished PS3.
If the PS3 boots up fine and you can start a game from the PS3 main menu, the PS3 motherboard is fine.