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; Each Kingdom Quest-line ending with the final showdown against the kingdom's Boss which is also the Kingdoms Legendary troop. Finishing a kingdom's quest line rewards the player with the Quest Giver that helped them through their travels (All Quest givers are Epic at their base rarity).
After completing the main Quests, you'll find that some Kingdoms still have the red notification icon as if there are new Quests to complete. This is because some Kingdom's have a Class Quest to complete after beating the boss!
Find more information on classes here: What Are Hero Classes?
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.