Assignments require you to delegate the work amongst your staff members. Assignments have different requirements, but they always need a department of a certain upgraded level and a corresponding staff type, with a certain amount of stamina in order to start it. Only staff that has not been assigned to a department or other assignment, can be used.
These missions take a varying amount of time to complete, depending on how intensive they are. Once you’ve started an assignment, all you need to do now is wait for the results. Remember that while your staff member is doing this assignment, it cannot be used for for other tasks, assignments, departments or be trained until the current assignment has been finished and the rewards claimed.