Penitently is how you do something when you feel sorry for a wrong and mean not to do it again: you look down penitently, speak penitently, smile penitently. It is the adverb of penitent, so it describes the manner, not the person. The word is rather formal and a little bookish; many people would just say sorry, or look ashamed. You hear it more in stories and careful writing than in the street. It is stronger than a quick “oops”: it says the person knows they were wrong and wants to be different, though of course a penitent look can also be put on, and listeners sometimes wonder if it is real.