I am going to disagree with the majority of the other answers and say this is possible, and I have done it quite a few times now.
There are quite a few different configurations - 20, 30 and 40 pin, then there are three usual subtypes where the connector is located in the top middle, bottom right or bottom left.
There are also some specialist ones for screens in ultra mobiles, ultra thin, long battery life etc.
I would first try plugging your laptop in to a HD TV if you have one, or just any 1080p monitor. If you can see output on it, your graphics chip is powerful enough for 1080p output. (may have problems if you want dual screens with another one though - can't rule it out).
If it is compatible, just pick a screen that has the cables in the correct place and type, then connect it.
Laptop LCD panels are very similar to computer screens in the fact that the monitor panel can "tell" the machine what modes it supports, and depending on your driver, the moment you configure it, it may go straight to 1080p.
FYI, this will obviously invalidate your warranty and not be that cheap, you may want to see if there is a 1080p laptop out the box that is better suited for yourself.