The raw versus JPEG debate is one of my favorite topics. As a software developer and photo nerd, I tend to gravitate towards the technical side of things.
This video by Steve Perry of Backcountry Gallery explains in easy-to-understand terms the advantages and disadvantages of each format. If you already shoot raw, you probably already know this information; however, the way Steve explains it may help you explain it to others.
Steve isn't a raw snob like some photographers are. He understands that the file format you choose to use depends on how you're going to use your photos. I've always said to use the tool that works best for you, and that also extends to file formats.
Using example photos, Steve shows what happens to each image when it is edited in Lightroom; however, this is not specific to just Lightroom. Comparing how the JPEG holds up with adjustments to shadows, highlights, white balance, and exposure, Steve shows just how a JPEG won't hold up to extensive editing.
This video is one of the better explanations of raw versus JPEG that I have seen. Some bonus footage at the end of the video also covers how Nikon and Sony users can set their camera to shoot raw and the advantages and disadvantages of 12-bit versus 14-bit raw files.