Story Quests are the quests that form the main storyline of a zone. In the leveling zones, these quests make up the Alliance Story; in the endgame zones, these quests develop the story of the zone in question. Story quests are the direct counterpoint of Side Quests.
The storyline quests follow on from each other in a chain that will take you from one end of the zone to the other if you follow them in order; however, it is possible to complete a number of quests in the chain before you even start some of the quests that theoretically come before those in story order. To indicate this, quests in the story chain will either have a “Previous Quest” or a “Prerequisite Quest” listed in the header (sometimes both). A Prerequisite Quest is one which must be completed before that quest can be started; a Previous Quest is one which comes directly before such a quest in the chain, but is not required for that quest to start.