Quests reward you for doing what you are told to do. Some quests require you to bring certain items while others require you to Restock an item or win a Battle against an opponent. There is no limit to the number of quests you can do. Most quests are timed and you will fail if you do not complete it on time. If you are asked for an item you cannot get for any reason, just wait for the quest time limit to run out of time and start a new one.
For more information about each quest, visit this Quest Guide. The Royal Fairy rewards you for completing Quests. You can also take part in the Weekly Quest Challenge.