The inverter board is a small (~1/2" x 4") board that usually sits on the bottom of the lcd screen and is encased in see through plastic. The purpose of this board is to change the low dc voltage supplied by your laptop into high voltage ac that is used by the ccfl (backlight). The ccfl is a very small fluorescent tube that is typically located at the bottom of the lcd, but above the inverter. It will be in a little tubular "nook" and will have two wires going to the inverter. Think of the combination as the ballast and lamp of an overhead fluorescent light.
If you are able to see an image on your laptop, even if it is very dark, and an image on an external monitor, your problem is generally a sign of the backlight (ccfl) not powering up all the way. This can be caused by (1) a faulty tube, or (2) a faulty inverter or both.
Since your screen was "half dark" then all dark, I think this is probably the issue - know how when a fluorescent starts to go out and one end turns black or doesn't light up?
If you decide to change this yourself (not very hard), you will need a soldering iron to connect the inverter to the ccfl. BE SURE TO DISCHARGE YOUR LAPTOP BEFORE MESSING WITH THE INVERTER - big time voltage across the terminals.
Take out the battery, remove charger, press power on for 5 seconds or so to discharge.