Missions are broken up into three categories: main missions that progress the story, sub-missions that can be obtained from various NPCs, and tasks from the towns' mission boards. Your current mission will be displayed on the upper-right corner of the screen when in the overworld, and you can view the Mission menu to confirm details about it...like the rewards for when it's finished!
You can check out mission boards in the slums and the Ark to take on tasks people have posted there. The boards are updated every morning at 8:00 AM, and missions may offer bonus rewards depending on what day it is.
ABANDONING MISSIONS
You can only have one mission of each type active at a time, so if you want to take on a new one, you will need to abandon the current one first. Abandoned missions can be accepted again at any time, though you will need to start them over from the beginning.