Quests are tasks and missions, either short or spanning a longer storyline, that are given to the player by the NPCs inhabiting Eorzea. As a reward for completion, the player can receive any combination of Experience points, Gil and items.
Quest NPCs are distinguishable by an exclamation mark above their head. Initiating conversation with them will tell you more about the quest, and let you accept it if you meet the level requirements. Upon accepting the quest, the current objective will appear in the Duty List on the right side of your screen. Further information is available in the player's Journal.
The various quests can be divided into the following categories.
Important quests used to tell the story of Eorzea and her people are called main scenario quests. These tasks are easily distinguishable from others by their unique icon. The story followed is a linear one, and often features voiced cutscenes. It will guide the character through the features of the game while constantly providing new challenges.
Each class in FFXIV have their own questline, with a new part unlocking at every five levels. The rewards are level-appropriate and class-specific gear. Some class quests are necessary to unlock further Jobs.
As with the Class Quests, the Job Quests are only available if the player is currently playing as the job in question. These also reward level-appropriate gear, and unlocks further skills.
All sidequests are optional, and can be done in any order, though the rewards will remain the same at any level. Sidequests can both be single quests or a short series of quests, and generally the only requirement for unlocking them is dependent on the player's level.
Somewhere in-between the main scenario quests and the sidequests, you find the side story quests. These are longer than the sidequests and can follow lengthy storylines. Examples are the Zodiac Weapon questline or the Anima weapon questline.
Chronicles of a New Era are in many ways similar to the side story quests, but generally demand more from the player. These are unlocked upon completion of the original main scenario quests from Final Fantasy XIV: A Realm Reborn and Final Fantasy XIV: Heavensward.
The Beast Tribe Quests are unlocked by completing a short introductory questline for the respective tribe. These allow you to increase your reputation with the tribe, eventually unlocking items from their NPCs. These can be completed once a day, using the 12 allowances that are granted daily.
Another type of repeatable quests, though these do not have a daily cooldown. You need allowances to do Levequests, which refresh at a rate of 3 allowances per 12 hours, with a cap of 100. These quests are sorted into three main groups, Battlecraft, Tradescraft and Fieldcraft, and will grant you Gil and Experience.