There are three types of Quests available for play.

[Main Quests]
The main story plays out in Main Quests.

[Sub Quests]
Sub Quests allow you to play through side stories.
Meeting the specific stage conditions will earn you special rewards set for each stage.

[Event Quests]
Event Quests are Quests with specific rewards such as items required for Reforging, or special Gear items.
Event Quests appear on certain dates or during certain times, so they are not always available. You must be a certain rank to participate in Event Quests.
Special Event Quests are also available during some time periods.