Fiend Quests are separate storylines, featuring unique puzzles that revolve around the Quest's hero.
Being separate from the main story levels, you do not need to complete the Fiend Quests to proceed with the main adventure.
You'll need to reach a specific level and meet certain requirements to unlock each of the Fiend Quests.
The requirements for unlocking a Fiend Quest are listed in-game when you tap on the Special Missions icon at the bottom right of the world map screen - just above your Fiendex. If you don't see that icon there yet, don't worry - keep on completing levels on the main map and it will appear there soon.