A fat bunny hug is a hug you give people when you are beyond drunk. You go up to someone, generally someone you've never met before, tell them you love them, puke all over their chest and then hug them. During the hug you squeeze them real tight so that the puke spreads all over each person's clothes.
Person one: Check it out, Ricky just gave that fat chick a fat bunny hug.
Person two: Oh shit dude that's bitchin'.