The following swords are listed by order of strength. Complete the indicated 
task to unlock the  sword.

Cutlass            : Default sword
Dutchman sword     : Complete the Pearl Vs Dutchman level.
Norrington sword   : Complete the Isla Cruces level.
Bone sword         : Complete the Davy Jones' Locker level.
Gun sword          : Complete the Sea Battle level.
Pirate King's sword: Complete the Shipwreck City level.
Davy Jones' sword  : Complete the Maelstrom level.
Ghost sword        : Complete the Hit Combo sub-mission on the Isla Cruces level.
Legend sword       : Complete the Hit Combo sub-mission on the Maelstrom level.