Quests are little missions you can do to get rewards. Some offer items you might need and others can unlock unique folks.
There are a standard set of quests that come with the game and additional quests that are added with downloadable content. Both types of quest are available at the pub but sometimes only during the day or night. Ellen and Keats both have different quests and some may start with one character and finish with another. Quests will only be playable when certain realms have been unlocked or certain portals have been reached.

Standard Quests

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 7

Downloaded Quests

The Kidnapped Folk Pack

The Bottom of the Sea Pack

The Alchemist Pack

The Origin of Belgae Pack

The Christmas Present Pack

The Fear in the Night Pack

The Visions of the Tower Pack