I'm going to throw a spanner in the works for the two above (although generally I agree and I also shoot RAW, it's not for everyone).
JPEG
When you shoot JPEG your camera does all sorts of processing to the image 'in camera' it will adjust the saturation of the colours, the sharpness etc etc.
This will give you nice crisp images at a set resolution that will look good as soon as you upload them onto your computer.
For most people this is fine as that's all they want, nice images from camera to computer.
If you decide to go down this route I suggest shooting at the maximum image size and highest JPEG quality your camera will allow.
The biggest problem with JPEG is the lack of 'editability' after you upload the image on to your computer.
If you're a Photoshop guru or you want an extra level of control over your images then JPEG probably isn't for you.
RAW
Just as JPEG does all sorts of trickery within the camera, RAW does non at all. It's a very pure representation of what you were photographing just like a film negative in old money. Just like a film negative, RAW requires a bit of processing in the darkroom (or in this case on the computer) to get the best out of the image.
The benefit of RAW is that it stores a lot of data that you don't actually see. Because of this you can change the exposure of the image, or the white balance (and a whole number of options) without losing too much of the image the way you would if you tried editing the JPEG.
However, almost every image will require some processing before it gets to a polished state. Also, because the RAW file stores extra data, the file size is considerably bigger than the same image in JPEG so if you have small memory cards or you're pushed for HDD space, RAW probably isn't your best option.
Conclusion
In conclusion then, if you want a rapid turn-around of image from camera to computer and probably online then JPEG might be the best option for you. If you're happy spending a bit of time getting the best out of each image using extra software such as Photoshop, Lightroom, RAWshooter or Aperture then shoot RAW.
Bear in mind that high quality JPEG will be fine for printing from the likes of Photobox, but if you're doing pre-press work you will need to convert them to CYMK .tiff. I would personally say that RAW is better for this but again it's down to turn-around time.
I've heard the argument that if you get the settings in the camera right in the first place you should never have a need for RAW but I disagree simply because I like the extra flexibility and control.
One thing you could try is shooting both for a while, I know this will give you two of every image and reduce the amount of images you get on your memory card but it will show you the difference between the two when they come straight from the camera and will also allow you to play about with the RAW whilst having a JPEG as a reference to compare your edited version with what the camera thinks is the best example of each image.
Hope that helps.
Panzer