Short answers, pulled from the story.
Radar was coined in 1940 by the United States Navy as an acronym for radio detection and ranging. The term later entered English and other languages as a common noun, losing its capital letters.
German inventor Christian Hülsmeyer was the first to use radio waves to detect distant metallic objects, demonstrating in 1904 that he could detect a ship in dense fog. He obtained a British patent on the 23rd of September 1904 for a full system he called a telemobiloscope. German military officials watched tests in Cologne and Rotterdam harbour and rejected it.
Radar measures distance by transmitting a short pulse and timing how long the reflection takes to return. The distance is one-half the round-trip time multiplied by the speed of the signal, with the half accounting for the trip out and back.
Chain Home was a network of aircraft detection and tracking stations installed along the East and South coasts of England before 1939. It gave the Royal Air Force vital advance warning during the Battle of Britain and fed the Dowding system that coordinated the response. The first five systems were operational by 1936, and by 1940 they stretched across the entire UK including Northern Ireland.
The Doppler effect is a shift in the frequency of the returning radio waves caused when a target moves toward or away from the transmitter. Only the radial component of velocity matters, so a reflector moving at right angles to the beam shows no relative velocity while one moving along the beam produces the maximum shift.
Modern radar is used for air and marine traffic control, weather monitoring of storms and tornadoes, ground-penetrating geological mapping, police speed measurement, and automotive adaptive cruise control and emergency braking. The first commercial radar fitted to aircraft was a 1938 Bell Lab unit on some United Air Lines planes.
The cavity magnetron was a key development in the UK that allowed relatively small radar systems with sub-meter resolution. Britain shared the technology with the United States during the 1940 Tizard Mission.