These are Quest that you need to complete in order to progress in the game. Completing Main Stories sometimes grants you access to the next area in the map or opens up Sub Stories and General quest.
Your current progression in the Main story will be always be pointed by the "NEXT" Quest indicator. This means that this is the quest that you need to complete before you can access the next area. Main Stories have a number before each title to indicate how far your progression is.
You can replay Story dialogues in your Drama Collection in the Sidebar menu.