Raw is the unprocessed image format that is recorded by your camera.
When you take a picture and save as a JPG you camera processes the image in accordance with the picture style you have chosen by default (standard, vivid, sepia and so on).
When you camera does this it makes a decision on how the final image will look and discards all of the information it doesn't need to achieve the same result.
The problem this presents is that if you want to use the information it has chopped, you can't as it is forever lost. So if you want to rescue an underexposed image or change the white balance or do some heavy processing, your options are limited.
Shooting in raw stores the image exactly as the camera "saw it" no post processing is done in camera. This means that all the information is present to be used - allowing you to pull detail out of blown highlights or lighten deep shadow.
This means that you initially notice two things about raw images - they are very big and they look flat, soft and lifeless.
In order to bring life to your raw files they need to be post processed using adobe raw, lightroom or the software supplied with your camera. Once processed you can save the finished image as jpg and retain the raw for processing in other ways.
So raw advantages:
Flexibility in processing
The ability to rescue images
Disadvantages
They are big
The need processing
They are manufacturer specific
JPG Advantages
Relatively Small File Size
Ability to use straight out of camera
JPG Disadvantages
Format is lossy and discards information, which gives more limited post processing options
As for which to choose - it really is up to you. My advice is that if space is a premium or you have total control over lighting and other inputs, jpg is great.
If you want total control over your images and are prepared to put some work in on PP, then the power of raw is unmatched.
Others will of course have different opinions, but I hope this helps a bit!
EDIT - this was a quick reply - so I may have missed some bits!