Plain English Meaning
When an aircraft flies fast enough, air speeds up as it flows over curved surfaces like the top of the wing. Even if the whole aircraft is below the speed of sound, the air over the wing can hit the speed of sound and form a small shock wave. These shock waves disturb the airflow and can cause buffeting, control problems, and a jump in drag.
Full Definition
Localized shock waves that form on parts of an aircraft when airflow over those areas reaches the speed of sound, even though the aircraft itself is still flying below the speed of sound. They occur in the transonic range, where some airflow is subsonic and some is supersonic at the same time, and they cause sudden changes in pressure, drag, and airflow behavior over the wings and control surfaces.