What does the "Reset Quests" option do?
This option will clear all progress you have made with the character you use it on. You will have to play the storyline again and unlock each quest before you will be able to choose to start at that quest.
What are the conditions to have the option available?
To have the option available, your character needs to have completed all quests, including killing Diablo.
If you decide to use the option on a specific character, you will have to complete all the quests again to be able to use it again.
What are the benefits of resetting quests?
By resetting your quests, you will re-enable first-time bonus experience & rewards for quests, boss kills and such.
As an example, killing Diablo for the first time currently guarantees a legendary drop for level 60 characters. This will be re-activated after having reset your quest progress.
To get the legendary to drop, the kill on Diablo must be the first since resetting all quests. Since loot drops per player, and not for all, playing in multiplayer guarantees the legendary for any player in the party that meets this requirement.
If you reset your quests then join a different game in act 4 and kill Diablo, you should get a legendary for killing him. However, since you didn't complete the first 3 acts you won't be able to reset quests again for another legendary. This means it's best to just go through all the acts after resetting. You will have to go through them all at some time or another anyways.
Best Answer
Bounty progress is not saved when you exit the game. If you want to get the reward for finishing all of the bounties in one act, you need to complete all 5 bounties without exiting the game.