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If a computer doesn't start, the on - off button probably isn't the problem.
If a desktop computer had been working and then won't start the most likely cause is a bad power supply. It's also possible that a RAM chip died. In either case, the malfunctioning part just needs to be replaced.
If a desktop computer is being assembled, check to make sure that all the components are properly seated.
For more ideas, see The Dead PC.
If the computer is a laptop, the motherboard probably is dead. A bad RAM chip is possible, but not probable. Just to play it safe, do swap out the memory. Another very unlikely cause is a malfunctioning battery. Try running the computer plugged in, but with the battery removed. Another possibility -- if you don't see any lights -- is that the connection for the AC adapter plugs is damaged.