There are millions of light sensitive 'photosites' (or pixels) on a digital camera sensor. When exposed to light, these sites record the brightness level of the light as a number. Normally, the camera sends all of these brightness values to it's internal image processor, which combines them with information such as the white balance, sharpness, etc.. The image processor converts all of this information into the final image, which is often stored as a JPG file on that flash card inside the camera.
If you shoot a picture as a RAW file, the camera skips all of the image processing steps. The information from the sensor is written to the camera's flash card exactly as it was recorded by the sensor. In order to view the image, the RAW file must be opened in a piece of software that performs RAW conversions. In this RAW conversion software, the photographer can make decisions about white balance, sharpening, and exposure, before the images is converted to a TIFF or JPG file. The RAW sensor data contains more levels of brightness information the the final JPG does, so it can be manipulated with less harm to image then if the image had been shot as a JPG originally.