You must earn 7 hearts from the governor's daughter of four different nations at the same time. You can view the number of hearts you have in the player's section in the menu -- it will always show the girls with the most hearts at the top.
This will be a similar process to getting married for each girl, minus the proposal.
When you meet a governor, he will likely introduce you to his daughter, who will be plain, attractive, or beautiful. She will either invite you to dance with her at a grand ball, or tell you once you reach a specified rank that you might be a suitable dance partner. Generally, the plainer the girl, the lower the rank requirement.
To get 7 hearts, you will need to follow the following steps:
1) Dance successfully
2) Give a gift (ruby ring, diamond necklace)
3) Visit – she’ll mention a jealous fiancé
4) Defeat the jealous fiancé
5) Dance successfully a second time
6) Governor tells you his daughter has been kidnapped
7) Rescue her and return her to her father
In order to do this for 4 different nations that are always warring with each other, you must be cautious in the way in which you choose to advance. Even after you have earned a rank from a nation, they can still place a bounty on your head which you can pay or attempt to work down by doing things that nation finds favorable. Using more of the non-confrontational advancement opportunities than the violent ones will make this much simpler.