Bonus Quests are stand-alone quests that invariably ask you to send a Sworn Sword to resolve a matter. Many of these will involve one of your Friends as a passive participant. These quests and their outcome have no influence on the story line or game progression, but they will give you Alignment choices.
Bonus quests have a relatively short duration (15-45 minutes, though some take notably longer) and show up intermittently in your quest log.
In general, they award a random amount of silver, 10 sworn sword training points and a random resource.