Quests are a new form of Challenge introduced into Fortnite: Battle Royale in Chapter 2: Season 5. They are the replacement for prior season's Weekly and Quick Challenges. Like most other aspects of the game, they are now classified by Rarity.
Each quest is offered by a different Character and can be seen in and out of games by checking the Quest menu. Quests come in Rarities from Common to Legendary, which each rarity offering a different reward to the player.
Legendary Quests are part of the replacement for Weekly Challenges, and come in the form of a 7 day timed Quest. Legendary quests are not unlimited, but the number of stages shown is not labelled. The first stage of a Legendary quest rewards the player with 55,000 XP and each subsequent stage gives 22,000 XP. Legendary quests allow other squad members to help with the challenge.
Epic Quests are the other part of the replacement for Weekly Challenges, or are related to the Battle Pass . These Quests come in numbered stages and either offer 20,000 XP, or a Battle Pass item.
Rare Quests are the replacement for Punch Cards. They are incremental goals for the Player to achieve that are shown when the Player gets close to completing them. These quests exist for the season, until incremental or full completion. They give 15,000 XP on completion of each stage.
Uncommon Quests are the replacement for Quick Challenges, specifically the daily ones. They are 24hr timed quests and offer 10,000 XP on completion. When completed they are replaced with a Common quest. You may only have 3 active at a time. These quests can repeat.
Common Quests are the replacement for Quick Challenges, specifically the repeated daily ones. They are 24hr timed quests and offer 1,000 XP on completion. When completed they are replaced with another Common quest, until the next day where they are replaced with an Uncommon quest. You may only have 3 active at a time. These quests can repeat.
Common quests are also the rarity given to quests started in game by interaction with characters. These quests last 1hr and require the player to do a variety of tasks, such as visit locations or forage items. Completing these quests rewards Bars instead of XP.
Per Match Quests
For more info on the subject, see Bounty (Quest)
Bounties are special quests which only exist in the match, and for a maximum of 6 minutes. The player's squad is tasked with eliminating an enemy player roughly shown to them on the map. If the enemy player is killed by your squad you get the maximum Bar reward, but if they are killed by another squad or die in another way, you get less bars and your bounty is 'poached'.
The Bounty also works the other way round, and shows if a player is set out to eliminate you or a squad member. If you survive the bounty, or your hunter's squad is eliminated, you will receive a bar reward, with surviving offering the most of the two.