FADEC (pronounced fay-deck) is an acronym, standing for Full Authority Digital Engine Control. It was introduced by NASA and Pratt & Whitney on an experimental F-111 in 1970, and today can be found in nearly every modern aircraft with the exception of some general aviation models.
In the early days of flight, engines in airplanes were controlled just like engines in cars. Mechanical controls, either direct or via cable or rod linkage, allowed the pilot to adjust the operating parameters of the engine. Fuel flow (throttle), mixture, carb heat and more were all managed by these methods - in some cases, when the engine was separated from the pilot by the length of the fuselage or by a section of wing, this meant more weight and more complexity as these physical links had to traverse long and indirect paths.
Especially as engines became more complex, with turbojet and turbofans not only requiring fuel flow control but on some high-performance aircraft required inlet and vane control as well, and with piston engines requiring the juggling of not only throttle and mixture but in some cases constant speed propeller controls, cowl flaps, electric fuel pumps, and more, the pilot workload increased. Any of these systems could fail, resulting in an in-flight emergency. Physical controls, furthermore, were vulnerable to lack of lubrication, to chafing in their hidden paths through airframes and wings.
The high performance and wide operating ranges of turbine engines were enough impetus to lead towards a solution. Analog electronic engine controls were tried, but in addition to being less precise, the large number of electronic noise sources inside modern airplanes made them more prone to misbehavior, especially without heavy shielding on their cabling. The NASA and Pratt and Whitney experiment mentioned above led to the first production digital computer for engine control, on the Pegasus engine manufactured by Rolls-Royce for use in the Hawker Siddeley Harrier VTOL fighter.
The "Full Authority" in the name indicates that a true FADEC does not have manual backups. The computer is the sole final controller of the engine's operation - the pilot tells the FADEC what he or she wants for performance, and the FADEC does whatever is necessary. Since computers are known to fail, production FADEC units tend to have multiple 'channels' - essentially, fully redundant sets of components to allow for failure. In some cases, multiple channels 'vote' on the action to be taken, in case one or more have malfunctioned in a more subtle manner than simply being unresponsive.
If a digital engine controller has manual bypass controls available, it's known as an ECU or EEC - Engine Control Unit or Electronic Engine Control - to distinguish it from a true FADEC.