Plain English Meaning
An airplane that has wheels you can pull up into the fuselage, wing flaps, and a propeller whose blade angle can be adjusted in flight. These extra systems make it more capable, but also give the pilot more to manage.
Full Definition
An airplane equipped with a retractable landing gear, flaps, and a controllable-pitch propeller; for seaplanes, an airplane with flaps and a controllable-pitch propeller. A single-engine airplane with a full authority digital engine control (FADEC) system meeting these performance characteristics is also considered complex for training and endorsement purposes.