Quests are gameplay-related tasks that you can complete to get rewards. The objectives range from popping bubbles of certain color to using certain characters on levels. You can complete these quests by playing any of the levels in the game. To view the available quests, open the Fun Stuff menu on the left side of the map screen.
The daily quest will change every day, even if you have not completed the quest. Make sure to complete it in time to get the reward. The two other quests change only once you have completed them, so you can take as much time as you wish to complete them.
In case you would like to skip a quest and get another one, you can do so with 3 coins. To skip a quest, tap the small trash can button next to the quest.