Both Bonus Quests and Fan Missions are a list of tasks that users can perform.
Bonus Quests are missions tailored towards artists - a list of objectives that, if reached, will net you some extra resources and rewards. These can be found in your Artist Hub.
Simply fulfill the necessary conditions listed in the Bonus Quest to be able to click them and collect your reward. See also: Help Article: "When does the Bonus Quests reset?".
Fan Missions are quests that you can complete to earn Stars for your favorite artists while watching their streams - it's a feature tailored towards the fans, not the artists themselves. Completed Fan Missions will automatically send Stars to the artists you're currently subscribed to, and you might receive new Mojis in your account.
Note that you must belong to artists' Crews to win them these Stars and that if you're an artist, you will not benefit your own artist account by fulfilling Fan Missions.