If someone speaks or acts conceitedly, they do it in a way that shows they think they are cleverer, more skilful, or better-looking than other people. The word is not kind: we use it when we disapprove of that attitude. It is the adverb from conceited, so it tells you HOW they did something — they smiled conceitedly, they talked conceitedly about their marks. Though actually, everyday talk does not use this adverb much. People are more likely to say “in a conceited way,” or “he sounded so full of himself.” You will meet conceitedly more in stories and slightly careful writing than in chat. It always carries a frown; nobody says it as a compliment.