It could be many things, if you have spare cables or different power/video/audio connections, move things around to see if that corrects the problem. The issue might be the system itself, which might have to be replaced.
Okay well I am guessing that its the first PlayStation and the light has not turned on when plugged into the power supply. 1) The cord... Yes you might have a dead power adapter replace it then try again. 2) the mother board is fried... these old systems like to over heat then die. Sorry it is lost.
If your Playstation does not always power on, be sure to check if all of the power cords are fully inserted. Check if the outlet is fully functional. If you are using a power strip or extension cord, check functionality by plugging a lamp into it. Also be sure to press the power button all the way down.