The main advantage of shooting in RAW is that you get to treat each shot as you want it to look and not as the camera decides it should look. When shooting JPEG files the camera carries out its own processing to create the JPEG image and information is compressed or can be lost altogether. This compression doesn't occur with RAW files (if it does it tends to be lossless compression) meaning images tend to be of a better quality. Having said that, the quality of JPEG images has improved greatly many camera models produce excellent quality JPEGs right out of camera.
RAW files give you more control over how your end image will look. You can easily adjust the exposure and tweak the white balance once back in front of your computer, something which is harder to do with JPEG files. In Lightroom you're also given access to sharpening and noise tools which can be used on RAW files. You don't have to worry about editing your RAW files either as you can always reset your adjustments if you don't like what you see. You can also go back to RAW files time and time again if you want to change things.
Downside Of RAW
Because RAW files are not compressed to the extent of JPEG files, the file sizes are much larger. This means they'll quickly eat up more memory and you won't be able to save as many RAW files as you can JPEGs. The camera also needs more time to process RAW files as there's more information to deal with which can slow the time it takes the camera to write images to the memory card. Having said that, it's always worth shooting in the highest quality setting you can as you can always down-size an image file but you can't do the opposite.
RAW files need to be processed and you'll either need special software to view the images or be using software such as Adobe Photoshop and Lightroom which can read RAW files. RAW files will show with different extensions depending on the brand of camera you're using for example, Pentax RAW files have a PEF extension.
Why Not Capture Both?
There is an option to capture both RAW & JPEG files on most cameras. Yes, it means space will be eaten up quickly on your memory card but it does give you more options and it also means you have a JPEG image you can view quickly or email off quickly if someone wants a quick check / proof etc. It also means that if you like using Art Filters the JPEG image is processed with the Art Filter applied while the RAW file remains as the original image which means you'll always have something to go back to if you decide you want to create a different look or don't want an Art Filter applied at all.