RAW is the raw information coming from the sensor. The camera turns this raw data into a photo (usually in the form of a jpeg file). The RAW file is basically a digital negative.
Jpegs have only 8 bit depth per color, but the sensors in the DSLRs tend to have 12 or more bits available. In making the jpeg the camera effectively loses all that information.That extra image information leads to several advantages.
The extra bits available in the RAW image give you more leeway in changing the exposure of the picture. Often you will see bright white areas in a jpeg (overexposed). A RAW file will (if it's not too badly overexposed) contain the information to repair that. A program like Adobe RAW is even that clever that it will still recover the brightness if one channel is overexposed.
Another advantage is the ability to finely repair the white balance of an image. I gues you know the principle, but in short, your eyes automatically adjust to bluish light (at dusk or in shadows), yellow light (evening sun) or even the greenish light of fluorescent tubes. Cameras try to do this too, but they often will get it wrong. In fact they rarely get it exactly right. In jpeg you can correct these errors too, but with a RAW images it's easier and looks better. A program like Adobe RAW easily lets you adjust a set of pictures to a single white balance (so they all look the same)
I think that is the biggest advantage actually. If you do a series of photo's and the colors don't match it just looks bad. Set them all to the same white balance in Adobe RAW and your done.
In essence RAW gives you more control over the image, but it can take a lot of time processing the images. In the distant past when people were still using film, a lab would develop your film and they would make the choice of how bright they thought the photo should be. RAW is basically a digital negative and now you have to set the exposure for every RAW image yourself. Of course auto settings can help, but I find I need to tweak them a lot to get the pictures the way I like them.
In some cases the camera software itself is bad. A PC is a lot more powerful than the processor in a camera and the camera has to be fast in developing the images so sometimes the camera just doesn't do a good job of developing the RAW image. I think the Pentax K10 was an example of this. That's rare though and certainly not the case with the leading brands. But still the camera does a lot of developing work for you that you personally might want to do different.
So in short, RAW contains a more detailed image and gives you more control over developing the picture. Obvious disadvantages are is that it costs time to develop the pictures and indeed that you need a bigger memory card (and slower continous shooting).
BTW people don't always agree on the advantages and disadvantages of RAW. Some will always shoot RAW and some always Jpeg and of course everything in between.
I bought a book about how to use RAW in Photoshop and I was hooked. Also read up on it through Google.
Ken Rockwell: JPG vs Raw: Get it Right the First Time He seems a bit biased against RAW, but I guess it's good to read both sides. His artcile does feel a bit dated though. He actually claims RAW does not contain more light info, but I'd say he's quite obviously proven wrong by a few simple tests. Also I think he misses the point that especially Canon oversharpens their images and a raw processor would by default not do that. A Canon Jpeg direct from the camera looks sharper, but does not contain more detail. Set the raw software to sharpen a bit more and you get the same picture.
PopPhoto: RAW vs. JPEG