If someone does something suavely, they do it in a smooth, charming, and confident way.
Doing something suavely can make people think the person is very cool, confident, and maybe a little bit fancy.
Examples:
He suavely complimented her on her dress.
She suavely glided across the dance floor.
The man suavely greeted everyone at the party.
The cat suavely walked into the room like it owned the place.
She suavely solved the puzzle in front of everyone.
He suavely convinced everyone to do what he wanted.