In the most base terms RAW files are almost like negatives of old film photography, they are a true capture of a scene before any processing work has been performed (most cameras will perform some in camera processing). These files contain far more detail than a processed image would (as odd as this sounds) and they therefore allow you to adjust things in major (or minimal) ways prior to the photograph being printed, in simple terms there is no loss of information from that captured on the cameras censor.
RAW files are not compressed so there is no loss of quality (as may occur with JPEGs etc), but what this does mean is that they can be large files (a typical 10 megapixel shot will be 10 megabytes, although this is not quite accurate), this can be something to bare in mind when using RAW as you need more space on a memory card.
The files are also unchangeable! When you edit a raw file you create a second file that accompanies the original and this has the changes, no real changes are made to the original file (something that can be used in courtrooms, where photographs may not be admissible, not that many of us would need this information).
RAW images are less sharp than Jpegs and often appear flatter in colour and contrast, this isn't an issue but this is due to the fact that there is a lot more detail in the dynamic range of a RAW file with more detail of both shadow and highlights having been captured.
Possible THE most important aspect to remember with RAW is that the file is not a picture file! And therefore cannot be printed or used online etc. as one would use a normal image file. So before these files are ready for actual use they have to be passed through what is known as a RAW editor/convertor (most cameras come with the manufacturers RAW processor that will be preset for use with your camera) These come in various forms from Stand alone editors to those packaged with the likes of Photoshop, a good list of software can be found at: http://www.fatphotographer.net/links/facts/software.html . Once these files have been edited in a RAW processor you will need to save them under another format (TIFF for uncompressed files or Jpeg for compression).
Hopefully you will now understand the basics of RAW, there are a lot of things that can be altered in RAW processors but they are for another article.