This is awarded for finishing the original (2.0) storyline. It won't finish until you're level 50, and it's several quests after that. Being an FF game, it has a fantastic storyline, and I wholeheartedly encourage you to follow it. Any quests released in further patches aren't required for the trophy.
If you already finished the storyline on PC, don't panic! All trophies are still obtainable. For this one, you just have to complete a single quest on PS3 if you've already finished the story. This can be an event quest, sidequest, class quest, or later patch story quest. The trophy will unlock upon hand-in.