Once you've been promoted your first rank for a given country, you'll be introduced to the Govenor's Daughter. She'll compliment you and ask that one day if your " such and such" rank, if you would mind taking her to the ball. Just continue on with your daily pirate activities, and once you've been promoted enough times to achieve that rank (or the rank after in a few cases) She'll barge in at your next meeting with the governor, and ask to dance. Just follow the on screen instructions on how to dance (basically just tap the circle when the outer ring touches the inner ring) and the achievement is yours.