If you are looking for a list of available quests, see Available quests
Quests can be done to get supply crates, special craft patterns and money. Go to your profile, then click the quest button. Quests are randomized every day based on your fishing skill, you can hold max 3 quests at a time, and hand in max 3 quests per day.
Quests you have handed in will not re-appear for 3 days.