RAW: is the raw image information as recorded by the camera (explained more fully in Wilt's post)
Jpeg: is an image file made from that data.
The choice of what to shoot is decided based on your wants and needs and understanding of the compromises involved in the choice you make.
Both file types have their distinct advantages, but the final decision is based on your wants and needs.
Bare in mind that every shot on your memory card starts life as RAW data in your camera.
The Jpeg is simply a result of allowing the camera to take that info and create an image file that can be read by anyone on any computer (one of Jpeg's few advantages, but a strong one to some)
The downside is that a jpeg can not contain all the image information your camera records, information that is available at the RAW level has to be reduced, and compressed to fit into that jpeg file.
First, large amounts of color information are immediately thrown out (most modern DLSRs record 14 bits per channel of color information, older ones 12. Jpeg can only store 8 bits of color info per channel)
Then the compression algorithms go to work, reducing image information further, and creating none image based artifacts from the compression process.
The differences in the resulting file types can be broken down in two categories to simplify.
1. Simple brute force Image quality.
RAW wins. It can not be argued. (some may argue to what degree RAW's superior image quality is visible, but this is a non argument, as obviously it is totally subjective. Different people with different needs has no bearing on the final fact that in every case the RAW files is superior in image quality )
2. Flexibility and workability.
Here the discussion is a valuable debate. Jpeg has smaller more portable Image size, and more "cooked" images due to allowing the camera to make processing decisions. It has universal viewing properties, and is the more useful file for those printing using an outside source. One can use a jpeg straight out of he camera if they are willing to make that compromise. RAW allows far more latitude to the shooter in post process, more room for recovery, and more flexibility in color, white balance etc. It can make the jpeg file for external use, as well as many other superior and equally universal file types like Tiff. It allows the user to control all aspects of the final image, leaving no decisions to the camera.
Most people get hung up on the latter, and seem forget that it's not the only reason to chose RAW. We see accusations of using it as a crutch. etc..
These thoughts do not take into account the simple concept that many simply want the best available to them.