I would suggest this following Canon camera. It's a little pricey at 219.99, but if you're looking for great video (it has HD) and good picture quality, this is what your looking for. This next price range down does not have HD video which means lower video quality. Most cameras have good photo quality to begin with, so this camera would be a great buy for a teenager.
Canon - PowerShot 14.1 Megapixel Compact Camera - Silver