Quests and Challenges are both found in individual kingdoms, each kingdom having its own specific set of them.
Quests are the main story or campaign arcs of the game. Each kingdom has its own quest line comprised of 20-30 battles to fight, ending with a final showdown with the kingdom's Legendary troop. Finishing a kingdom's quest line rewards the player with an Epic troop (the guiding character or quest giver who has been with the player through the quests).
Challenges are an additional set of battles that are unlocked by reaching milestones in the quest line. Each kingdom has 7 challenges with 5 levels of difficulty. Completing challenges rewards the player with additional Souls, scaling with the difficulty of the challenges.