In shortA RAW file records the output of all the tiny cells on your camera's sensor when the shot was captured, with each cell producing a brightness value for the amount of red, blue and green light falling on it.
In a sense, it's much the same for film cameras. When you take a shot, what's on the film? Just the chemical gel, with some of the light-sensitive chemicals changed by their reaction to the red, blue and green light falling on them. You then take the film to be processed to produce a negative -- a one way process.
Without RAW, your camera immediately converts the sensor output to jpeg, before storing it on your memory card -- a one way process. And because the sensor captures far more information than can be stored in jpeg format, a lot is discarded during the conversion.
What is the extra information that is lost?The range from dark to bright recorded by the sensor is much wider than can be held in a jpeg; in a RAW file, it's like you have 7 different exposures in one, from -3 stops to +3 stops. Say you took the shot at 100 ASA, f8, 1/100s, then most digital cameras' sensors will capture the equivalent of f22, f16, f11, f8, f5.6, f4, f2.8 all in one RAW file.
Of course, with a jpeg, it records things at just one exposure setting, the one you (or the camera) set when you took the shot.
What other practical things does a RAW file allow you to do?
White BalanceThis is real useful. You see, another thing that conversion fixes in stone is the "white balance". Unless you remembered to set it correctly, shots taken indoors or under street lamps can end up with a horrid yellow or blue colour cast. But with RAW you get to play with the white balance (also called colour temperature), try it this way, try it that, to see what works best for the shot.
SharpeningEven when perfectly focused, all sensors produce a slightly unsharp image and your camera, in converting to jpeg, applies a standard amount of sharpening. Well, you guessed, a RAW file doesn't have any sharpening applied, it's something you get a chance to play with once it reaches your pc.
Downside to shooting in RAWThe size of a RAW file is much bigger than jpeg and therefore fills up your memory card much quicker, and also takes longer to save to it. It also means it takes longer for you to end up with a finished image, as you have to make all the decisions (after playing around!) rather than the camera making them for you at the time of the shot.
To sum upRAW is like an exposed but unprocessed film.
Shooting in RAW leaves all the important decisions to you and, maybe most importantly, gives you a lot of flexibility to correct exposure and white balance where the camera (or you!) gets it wrong.
As usual, your comments and questions are welcome :)