Also known as the circular file, and more commonly, the trash can, the fuck it bucket is the place in which rejected homework assigments and other things that one wishes not to deal with are thrown. Fuck it bucket can also be used in a less literal sense to show that someone is giving up on something.
The "fuck it" in fuck it bucket comes from the term for which it is named which means "forget this" or "screw it."
I didn't feel like doing my homework last night, so I tossed it in the fuck it bucket and called it good.
Person 1: How's that new song coming along?
Person 2: Sucks. I think I'll throw this one in the fuck it bucket.