The most common way that a computer "won't turn on" is when the PC actually does power on but does not 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.
Test your monitor. Before you begin more complicated and time consuming troubleshooting with the rest of your computer, make sure your monitor is working properly.
It is possible that your computer is working fine and your monitor is your only problem.
Verify that your PC has fully power cycled. In other words, make sure your computer has completely restarted - ensure that it is coming on from a completely powered off state.
Often times a computer will appear to "not be on" when actually it is having problems resuming from either Standby/Sleep or Hibernate power saving mode in Windows.
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.