Bandwidth theft or "hotlinking" is direct linking to a web site's files (images, video, etc.). An example would be using an img tag to display a JPEG image you found on someone else's web page so it will appear on your own site, eBay auction listing, weblog, forum message post, etc.
Bandwidth refers to the amount of data transferred from a web site to a user's computer. When you view a web page, you are using that site's bandwidth to display the files. Since web hosts charge based on the amount of data transferred, bandwidth is an issue. If a site is over its monthly bandwidth, it's billed for the extra data or taken offline.
A simple analogy for bandwidth theft: Imagine a random stranger plugging into your electrical outlets, using your electricity without your consent, and you paying for it.