Quests are randomly-generated procedural tasks with varying story, creatures, and places. Unlike missions, Quests focus more on multiple objectives as opposed to solely slaying a boss. Quests are not yet implemented.
There are also Missions, which are marked by a Crossed Sword symbol on your map. The current main objective of a Mission is to defeat an enemy Boss, who is often located within a fortified structure surrounded by hordes of smaller enemies.