Piaget's theory on adolescent cognitive development is particularly complex, and so out of brevity this answer will restrict itself to one area which is that of mental operations. What happens to adolescents as they enter this stage of life, according to Piaget, is that they begin to be able to think in a more abstract fashion rather than merely in concrete fashion. Piaget termed with "formal operations," which he used to refer to the ability to perform mental operations, such as predicting what would happen next, but being able to incorporate abstract concepts such as "justice" or "morality." In addition, adolescents are able to talk about and describe these concepts. Teenagers are known for developing their own causes at this point in life, and formal operations is the name that Piaget gave to this cognitive developmental process.
The big shift therefore is from children who are only able to talk about what they know and concrete situations that they have observed and can predict, to teenagers who are able to talk with confidence about abstract situations which they have never personally undergone. This can be particularly linked to empathy, as teenagers are able to imagine what life would have been like if they had lived in a certain period of history as a certain person. For example, a teenager might be able to imagine what life was like as a slave in the South in the years before the Civil War, drawing on their understanding of concepts such as justice and morality to help them explore this further.
This is just one of the five changes that Piaget said adolescents underwent as they entered this period of life. Piaget's theories of adolescent cognitive development still remain some of the most important in this field, and even though his work has attracted a lot of criticism, at the same time, their continuing popularity indicates their importance.