Plain English Meaning
A BATD is a simple desktop-style flight simulator that the FAA has officially approved for use in pilot training. It looks and behaves enough like a real cockpit that pilots can practice procedures on the ground, and some of that practice can count toward training requirements.
Full Definition
Basic Aviation Training Devices (BATDs) are ground-based training devices approved by the FAA that replicate the instruments, controls, and basic flight characteristics of an aircraft. They are used for procedural practice, instrument task training, and a limited amount of loggable flight time toward certain pilot certificates and requirements, as defined in the FAA's for the device.