Someone who acts sullenly seems gloomy, dissatisfied — not happy to be there. If you act sullenly at your grandparents' anniversary party, you can expect to be taken aside and told, "Cheer up. Now."
The picture of someone acting sullenly? It's a person slumped in a chair, arms crossed, sighing a lot. The adverb sullenly is used to describe something that's done glumly, or in a very bad-tempered way. Anyone who is in the mood to complain will speak or respond sullenly, like a little kid who's been dragged along on a car trip and speaks up sullenly from time to time to ask, "Are we there yet?"