The most common way that a computer "won't turn on" is when the PC actually does power on but doesn't display anything on the monitor.
You see lights on the computer case, probably hear fans running from inside, and may even hear sounds, but nothing at all shows up on your screen.
There are several possible reasons why your monitor isn't displaying information, so it's very important that you step through an ordered process like the one outlined here.
Important: If your computer is, in fact, showing information on the monitor, but is still not booting fully, see How to Fix a Computer That Won't Turn On for a better troubleshooting guide.
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Tips & More Information
Are you troubleshooting this issue on a computer that you've just built? If so, triple check your configuration! There is a very, very good chance that your computer is not booting up completely due to a misconfiguration and not an actual hardware failure or other problem.
Did I miss a troubleshooting step that helped you (or might help someone else) fix a computer that's not showing anything on screen? Let me know and I'd be happy to include the information here.
Is your computer still not showing anything on the monitor? See Get More Help for information about contacting me on social networks or via email, posting on tech support forums, and more.