If possible try the external monitor port with a working monitor....the port will either be on the side or on the back of your laptop.
Make sure the second monitor is getting a signal....most will say something like 'No Signal Input" if it isn't.....most laptops use Fn + F5 to scroll through the output options of the second monitor but you may have to consult your manual on that one.
If you can replicate the problem and the second monitor stays on then we can assume that the problem lies within the screen itself and is not software related.....if the second monitor goes off too then it will be a software issue rather than a hardware problem.
If you find the second monitor is ok and stays on, try reducing the brightness of the laptop screen to minimal....again this is usually a Fn + key combination.
If you can then turn OFF the screen backlight (yet another Fn + key combo) and then back ON again, and the screen comes ON, if a bit dim, then the problem lies with either the CCFL backlight tube or the inverter that powers it.
Inverters are expensive but very easy to change, CCFL tubes are very cheap but a right pain to fit...they are extremely fragile.
A way to test if this IS the problem is to slowly increase the brightness of the screen (if it works at all) and see if it goes off again at a certain brightness....this is a sure way to tell if it's backlight trouble (or inverter).
If the second monitor goes off too then it's more likely a software issue and this could be any number of things. The power management has already been mentioned and would 90% of the time be spot on...but the fact that the screen doesn't come back on with a key-stroke or mouse-move seems to indicate it may be a little more deep-rooted.
Try the second monitor first and see if it gives you any indication.
Regards.