A Quest is a series of fights which a team of heroes needs to complete as a continous serie. During a Quest, HP is not refreshed when moving to the next fight, so any damage taken in previous fights is lost for the current and any remaining fights.
After completing a Quest, you will be rewarded a jewel. You can then replay the same Quest (which will have tougher Opponents); if you complete the Quest again, the jewel will be upgraded. This can be done up till 9 times for each jewel.