It's "I feel bad."
That's because "feel" is a linking, or connecting, verb. As its name indicates, a linking verb connects the subject with the adjective.
"I feel badly" means I'm very bad at feeling things. I run my fingers across the class and do a terrible job of it.
"I feel bad" means just that--I'm upset, sad, disappointed.
But, don't feel bad; lots of people get this one wrong.
