I would check the on/off button first, but this usually involves dismantling the unit. Check for continuity using a multimeter on the contacts on the on/off switch with machine unplugged from the mains and battery taken out (if its a laptop).If continuity is ok then you need to see if power is getting to that switch. Check continuity of mains lead after unplugging from the wall, or if its a laptop power supply unit, check that it is outputting power from the connector that goes into the laptop.
Good Luck.
CJ, March 2007