The Challenge system is based on daily quests, most often associated with a Dungeon or Raid instance with their respective Daily Quest. These reset and change every day at 4AM Server Time. Completing said Challenge can reward you with such treasures such as XP, upgrade materials, and a special currency called Solar Energy to purchase rewards from the Dragon Express.
There are two kinds of Challenges that a player can partake in:
Daily Challenge is constructed of many different daily challenges based on completing a Daily Quest. Typically, completing 3, 5, and 7 quests will grant you rewards, although certain events include a reward after completing 2 quests as well. Daily Quests resets at 6AM Server Time.
Weekly Challenge works in the same way, but is mainly finished by doing Raids and provides higher rewards than the Daily Challenge. As you might have guessed on its name, the Weekly Challenge can be done once every week and resets on Wednesday at 6AM Server Time.
For a full list of the available Challenges, please see this article's subpage.