Quest Journal

Each pirate carries a journal or book to keep notes of their current goals. When they encounter and accept a new quest, it is placed in the book. As the pirate completes the quest, a running percentage informs them of how far along into the quest that they're currently doing. Basically just their progress, and what they have to do, where they have to do it, and sometimes, what they have to kill or steal, etc. This journal is given to the pirate by Will Turner at the beginning of the game.

Some quests automatically appear in the journal once a pirate reaches a certain Notoriety level. These quests are often for new weapons, treasures or teleportation totems.

A pirate can get assistance in completing a quest, by selecting it with the Track option. Then, their Compass will indicate the direction where the quest object is, much like Jack Sparrow's compass, and it is wondered by many if it actually is Jack's compass. Also, a tracked quest object will place a mark on the pirate's Map when they view it.

Story Quests

These quests are part of the main quest given to you, usually by main characters and involve advancing the storyline. As a reward, you will receive Notoriety points.

So for there are only 3 story quests in the game:

Fortune Quests

These quests are optional and usually given to you by townsperson or by main characters in addition to the fame quests. The reward is usually gold, treasure, or Notoriety.