This question is fairly complex.
The first operational turbojet engine was built by Sir Frank Whittle and was operated in April 1937 for the first time, several months before Otto Von Hain's less practical engine. Despite this, the Germans were the first actually in the air with a turbojet powered aircraft, simply by installing a jet into an airframe: the He-178.
The famous Messerschmitt Me-262 was the first fighter jet to fly in prototype, though Britain's Gloster Meteor actually entered service first by a matter of days. These were the first practical jet fighters. Germany also operated an Arado twin jet bomber/recon aircraft.
Despite being less aerodynamically advanced, the Meteor enjoyed a long operational career in various forms. The Me-262 was hampered by unreliable engines and many other issues, not least of which was the inevitable defeat in WW2. The 262 inspired much of the former allies jet design for the next decade.