When we think of psychosocial development and psychosocial tasks, we usually think of Erikson and his stages of psychosocial development.
In this theory, human development is defined as going through 8 stages from infancy until adulthood. The psychosocial tasks are the developmental markers that define each stage. They are social and physical challenges that the person is able accomplish, thereby marking which stage he or she has mastered, and which he or she is in at any given point in life.
For example, in the infancy stage, the only psychosocial task the baby is expected to master is to learn whether caregivers actually provide basic needs or neglect them. In this way, the baby comes to rely on and cling to only those who meet his or her basic needs. In contrast, during the "care" stage (between 45 and 60 years old) a psychosocial task includes making meaningful contributions to society and actively furthering the progress of future generations.