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.
Automatically Given Quests
Each quest is offered by a different Character with a bit of flavor text, and can be seen in and out of games by checking the Quest menu. Quests come in Rarities from Common to Legendary, with each rarity offering a different amount of XP to the player.
Legendary Quests are part of the replacement for Weekly Challenges, include story-driven quests, and come in the form of a 7-day timed Quest series. Legendary quests are not unlimited, and the number of stages is indicated in the completion requirements (usually in the form of "Stage X of Y"). The first stage of a Legendary quest rewards the player with 45,000 XP and each subsequent stage gives 30,000 XP.
Epic Quests are the other part of the replacement for Weekly Challenges, are story-driven quests, or are related to the Battle Pass (also known as "Special Quests"). These Quests usually come in numbered stages and either offer 24,000-30,000 XP (depending on stage), or a Battle Pass item. Some quests include shared progression among squadmates (meaning your squadmates can help you complete that quest, and you can help complete theirs).
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 12,000 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 17,000 XP on completion. When completed they are replaced with a random 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 1,700 XP on completion. When completed they are replaced with another Common quest, until the next day when they are replaced with a new Uncommon quest. You may only have 3 active at a time. These quests can repeat.
In-Round Initiated Quests
Bars Quests are started in-game by interaction with characters or payphones during a match, and can be of Common, Uncommon, or Rare rarity, depending on their difficulty. These quests usually either last 1 hour or only exist in the match, and require the player to do a variety of tasks, such as visiting locations, crafting weapons, or foraging items. Completing these quests rewards Bars instead of XP.
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 and/or their 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 considered 'poached' (although this still counts as 'completing' a bounty).
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 and/or their squad is eliminated, you will receive a bar reward, with surviving the time limit offering the most of the two.