Germany's Messerschmitt Me 262 was the first jet powered aircraft to be used operationally. It was designed before the war, but due to engine development problems, the aircraft was not put into service until beyond mid 1944. By this time the Allies had entered Europe and much of Germany's Luftwaffe was spent as an effective force.
The sudden appearance of this fast jet, however, did catch the allies off guard for a while and the Me 262 was responsible for 509 kills.
To combat this fast aircraft, the allies used ground attack fighter bombers to destroy them on the ground or taking off and landing. Fortunately for the allies this excellent jet fighter entered the war too late, there was not enough of them and fuel availability all took their toll on the over all effectiveness of the craft.