If the phone is not powering on, first make sure that the phone doesn't have a completely drained battery:
After you have confirmed the phone is charged, try to power it on by pressing and holding the POWER button for 10-15 seconds.
If the phone still doesn't power on, holding down certain hardware buttons can force the phone to reboot. See How do I reboot the phone using hardware buttons? for details.
Note: Some phone models may not have a notification LED.