At the very start of the movie the young Miss Swan found young Will Turner floating in the ocean. When she went up to look after him she found a pirates necklace and took it knowing that if her father saw it he would do something to him.
Many years later a crew of pirates come and invade Port Royal. They take Princess Swann thinking that she was the one they needed for their plot.
Captain Jack Sparrow and Will Turner team up to go save Miss Swann. Also when Barbossa realises that Elizabeth isn't who she says she is . Meaning that they have the wrong blood to end the curse.
When Babossa let Miss Swann and Jack go in exchange for Will. They exchange people but then after a few days on an island they were dropped off at by Babossa and his crew they set off to try and save will and kill Barbossa.
When Jack and Will kill Barbossa and end the curse. Also when they save Miss Swann and return back to Port Royal.
Miss Swann declines Commodore Norringtons proposal and gets her ins with Will. Jack Sparrow becomes captain of the Black Pearl again.