The paper that is wrapped around over half of your burger that makes eating the burger harder, but keeps the lettuce and sauce from falling onto your tray. (almost resembles a real diaper in the sense that you use it it's early stages to collect the released exrcement.) Once your burger is about 3/4 of the way done, it should be out of it's burger diaper(s) and be held so that the uneaten side is facing up so that it may be finished. Some places that give the burger diapers are In n' out, The Habit, etc...
"Man, I really hate having to move the burger diaper down every time I want a big bite. But these burgers are so sloppy, I can't just take it off. That would only end in tears..."