As long as the system itself is okay (booting, etc.), could be the cable to the monitor is broken.

 

My money's on Zamo's explaination. It's a very common fault with laptops.

To find out for certain, shine a bright desk lamp or a torch onto the screen while it's turned on. If you can see a faint picture hiding in there then that will tell you that the LCD panel and the rest of the laptop is working fine, but the screen backlight behind it isn't.

If that's the case, it's a guessing game as to which part of the backlight assembly has gone, but the most common is that the inverter board that provides power to the backlight has snuffed it. Either way, it's a hardware repair job I'm afraid. If it's under warranty send it off to Sony, otherwise if you know what you're doing or know someone who does then google around for information about replacing inverter boards. Most computer repair shops charge ridiculous fees to repair laptop screens for you but that might be your only option. If doing it yourself, you'll probably have to read the part number off of the board that's in there now to find a match. The price of the part varies depending on the age of the laptop, if you can find one at all. (I couldn't find one for mine, spent 3 years running off an external monitor before I caught a cheap second hand one on EBay).

Otherwise, running off of an external monitor should work just fine. As long as portability isn't an issue you get used to it.

 

press Fn on your keyboard along with the contrast button