Bad is an adjective. It describes nouns or pronouns. It is often used with descriptive linking verbs like look, feel, sound, or to be.
Incorrect: She felt badly about missing the date.
Correct: She felt bad about missing the date.
(Bad describes the pronoun she.)
Incorrect: Things looked badly for the Mudville nine.
Correct: Things looked bad for the Mudville nine.
(Bad describes the noun Things.)
Badly is an adverb (like well). It describes verbs and should be used with all verbs other than linking verbs.
As many adverbs do, it usually answers the question "How?"
Incorrect: Mudville played bad last night.
Correct: Mudville played badly last night.
(Badly describes the verb played.)
See also Irregular Comparisons