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 and hear fans running from inside 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 I've outlined here.
1. Test your monitor. Before you begin more complicated and time consuming troubleshooting with the rest of your computer, make sure you monitor is working properly.
It's very possible that your computer is working fine and your monitor is your only problem.
1a. Ensure your monitor has power. Do you see any lights working on the front, usually flashing amber or green.
If you do see lights and there is power...
1b. Check the connections that display lead is plugged in correctly and is not loose. It is probably worth disconnecting and reconnecting the lead.
1c. Swap it with another monitor near by. Does the new monitor work, does the old monitor work on another PC, if it is the latter then you have broken monitor and needs replacing.
2. Verify that your PC has fully power cycled. In other words, make sure your computer has completely reset - make sure that it's coming on from a completely powered-off state.
Often times a computer will appear to "not be on" when actually it's just having problems resuming from either the Standby/Sleep or Hibernate power saving mode in Windows.
Note: You can power off your computer completely while in a power saving mode by holding the power button down for 3 to 5 seconds. After the power is completely off, turn on your PC and test to see if it will boot normally.
Unplug and reattach your keyboard and mouse as well. There isn't a great possibility that the keyboard or mouse is causing your computer to not turn on fully but we might as well reconnect them.
3. Finally, if all else fails, you'll probably need to seek professional help.