Assuming a separate monitor (as opposed to a notebook screen), is it safe to assume that you have checked that the monitor is (a) plugged in to the power source, (b) connected to the video output of the PC, (c) is turned on, and (d) its power-on indicator light (if any) turns on?
If yes to all of the above, you might want to connect the monitor to another computer to verify whether or not it is defective. If it works OK, then the next thing I would try would be to turn off and unplug the computer, open it, pull the video card, make sure it's not covered in dust, then re-seat it and reconnect the video cable (maybe you'll be lucky and it's a case of the card not being securely connected to the motherboard or a loose video cable).
If that doesn't help, then it's find another video cable and try that, in case it's a case of a bad cable.
If that doesn't help, you could try the video card in another PC that you know works.
At that point, if you've determined that the monitor, video card, and video cable are all working, then it's probably something wrong with PC itself, at which point you're out of my league.