bad, badly

To smell bad means to stink.
To small badly means to mistake garbage for roses--or roses for garbage.

Bad is an adjective--it modifies a noun (thing or idea).
Badly is an adverb--it modifies a verb (an action of some kind, even a mental action, like thinking ).

To feel bad means to be sick or to hurt. Saying it this way means the person is what feels the bad feeling.
To feel badly means to grab a cactus when you meant to grope for a towel. Ouch. Here you have a person doing a bad job of feeling.