Quests are little mini-challenges and additional tasks that reward you as you play.
Quests will appear at regular intervals and they'll set a goal which you can monitor over time. They'll include tasks like collecting a specific number of candies or to reach a certain level number, for example.
You can check your current quests at any time in the game by tapping the checklist icon at the bottom of the map screen. Quests will reset after a period of time, so make sure to note the timer at the top of the quests log to make sure you have chance to complete them in time.
If you have a quest that you don't want to take, or you're finding it difficult to complete - don't worry! Just tap the 'trashcan' icon in the top right corner of any given quest to be assigned another.