Think of raw as a negative in the sense that its ALL of the information contained in the shot but no decisions have been made yet about how to process it. A jpeg is a file that's been processed from that raw file with various settings applied to make it look a certain way - think of it as a print. Every camera shoots raw, but a lot of cameras immediately convert the file to a jpeg (based on the settings that are built into the camera and some that you can modify) and junk the raw file - these cameras don't offer you the option of a raw file even though the file originated as raw data. So, if you have a camera that allows you to choose which format you want (any reasonably nice camera, basically), if you choose the raw file, you get an unprocessed file and you can make ALL of the processing decisions. If you choose to get a jpeg file, you get a processed file with most of the decisions made already - you can still tweak a jpeg file in an editor, but you don't have anywhere near the latitude or the amount of data available to you that you'd have in the raw file.
This description makes the process of working with raw files sound cumbersome, but it really isn't with today's soup to nuts software (Aperture and Lightroom being the most popular examples) that will process your raw files for you upon import and save them, giving you the ability to work on them additionally in any way you want. You can modify the settings it uses when it imports the files to make them more to your liking if you're not thrilled with the default settings (which generally try to look reasonably close to the manufacturers jpegs). Importing a huge batch of raw files takes a little bit longer than importing a huge batch of jpegs, but there's not much downside to using raw beyond that. The upside is that if you want to try to pull some detail out of the shadows or push down some of the highlights, change the color saturation or the contrast curves, etc, ALL of the original information is still there to work with. With a jpeg, a lot of the decisions have been made and the extraneous data has been tossed aside (which is why jpeg files are notably smaller than raw files). You still have some ability to tweak jpeg files, but not nearly the control you have with raw. If you work with raw files and Aperture or Lightroom are holding onto your raw files, when you've made all of the edits to them and want to print them or post them on the net, send them to friends, etc, the program THEN exports that version of the image to a jpeg for distribution, but it still holds on to the raw file and all of the data, so you can continue to make changes or new versions of a shot with all of the orignal information intact.
A lot of camera manufacturers produce really nice out of camera jpegs (with my Fuji X100 they're so nice I can't seem to beat, or even equal, them with my raw processor, so I tend to shoot jpegs with that camera). With most cameras, though, I can do a better job, or a job more to my liking, with a raw file than with a jpeg. If you have no interest in processing your shots beyond what comes out of the camera, there's not much reason to shoot raw (other than the possibility that you might change your mind someday). And shooting jpegs generally allows the camera to write the file to the card more quickly and get on with other operations, in addition to being able to fit FAR more jpeg shots on a given SD card than raw files. OTOH, if you think you even MIGHT want to delve into processing, raw has some real advantages most of the time. And very little downside given how inexpensive storage has become and how fast most modern cameras are...
I hope this helps as a basic explanation...
-Ray