What was the first fighter plane used?

The first aircraft actually designed to shoot other aircraft down was Germany's Fokker Eindekker, around 1915. The British and French were also working on such projects that soon came to fruition.

At the beginning of WW1, planes were first starting to be used by the military of all sides. Popular imagination saw the aeroplane being used as a bomber (HG Wells 'The Shape of Things to Come' was written before the Wright brothers first flight but predicted cities flattened by aerial bombardment). The poor performance of early aircraft meant that bomb loads were small and at the outbreak of WWI the only practical use was an observation platform especially for artillery spotting. It was noted that artillery fire became much more accurate and effective when aeroplanes were used to spot. It became important therefore to send up other aircraft to remove them. Thus the first 'fighters' were born. At first, these aircraft were similar observation aircraft but pilots would use pistols, rifles grenades and machine guns with varying effects even grappling hooks were tried. The French Morane-Saulnier monoplane, armed with forward firing machine guns through the propeller (protected by Armour plates) was introduced in April 1915. This innovation made the scouting aircraft a more effective fighter.

However, probably the first true fighter (i.e. designed only to shoot down other aircraft) was the German Fokker E1 monoplane with synchronized forward firing machine guns (the guns did not fire when the propeller was in front of the muzzle). This was introduced later that summer. The British adapted the De Havilland DH2 and the Royal Aircraft Factory FE2 with some success later still. The British designs used a "pusher engine" which meant the engine and propeller was at the back of the aircraft. This allowed the pilot or a 2nd gunner to fire a machine gun forward without being obstructed by the propel.They made the fighter plane to fight the Germans on world war 1.