Quests are gained by talking to the quest-giving NPC, who is marked with an exclamation mark above their heads, and are finished by completing the conditions that were laid out at the beginning of the quest. The NPC will have a question mark above their heads when the quest is finished. You can accept more than one quest, and can have a number of 'active' quests at any one time.
The Quest Log(or the Quest Page) is a record of all the quests that you currently have active. On this page your active quests are displayed in a list down the left hand side of the page, while the completion details of the selected quests are shown on the right. The quest stages are marked with a tick once they are completed, allowing you to chart your progress throughout the quest.
Ability Statue Quests Coming soon.