The backlight for the LCD screen in most laptops is a cold-cathode fluorescent lamp, which is basically just a small fluorescent tube. Just like fluorescent room lighting, they eventually burn out. Unlike room lighting, they are not made to be replaceable.
Here's my quick and dirty (and cheap) project to replace the lamp on my old laptop, who's backlight burnt out recently. I expect you could make this look a lot better if you were willing to put some more money into sourcing parts.
Materials: Screwdrivers, Dremel, electrical tape, wire crimping tubes.
The specifics of disassembling any particular laptop are going to vary widely, so I'm going to skip that step for simplicity. If you're not confident just diving in to your laptop, you'll frequently be able to find illustrated parts breakdows (IPBs) at the manufacturer's site for some guidance.
NOTE: High resolution photos available on Flickr: http://flickr.com/photos/killerrobotclan/sets/72157605872353185/detail/