A subquest is like a smaller objective within the quest you are embarking on that has its own smaller cash and material reward pool. They can range from anything to slaying a certain number of specific small minions to breaking the certain part of a large monster. If you can, you should always target to complete the subquest along with the main quest for additional money and materials. Check Quest Info when out in the area from the menu to see the subquest. When selecting quests from the village gals, subquests will be displayed below the main objective. Not all quests have a subquest.
Additionally, if you complete the subquest, you can go to the menu and select the End Via Subquest option. This will only give you the subquest’s monetary and material reward but it is a great way to either speed farm a particular quest for a particular material, or bail out early if the hunt isn’t going well so that you don’t outright fail.
If you haven’t completed the quest, and the first time you take it, you end via subquest, the quest will not be marked as completed. Completing a quest for the first time always requires you complete the main objective.