"Bucket" is a hip-hop word used in New York to describe an oversized fitted cap that's worn as low over the eyes as possible. Joe Budden, who's pictured above is a good example of someone who always rocks his "bucket" low. On Jay-Z's "Blueprint" album he opens the song All I Need with the words:
"My gear is right... (check)
My bucket is low... (check)
My Rocawear is fittin' incredible..."
Also on the Diplomats album "Diplomatic Immunity" Cam'ron says on the song I Really Mean It:
"Hey yo lock my garage, rock my massage
F*** it, bucket by Osh Kosh Bgosh
Golly I'm gully, look at his galoshes"