Short answers, pulled from the story.
A fighter aircraft is a military aircraft designed primarily for air-to-air combat, and its main role is to establish air superiority over the battlespace. Controlling the airspace lets bombers and attack aircraft strike enemy targets while preventing the enemy from doing the same. Many modern fighters also have secondary capabilities such as ground attack.
Early fighters needed a forward-firing gun because the best direction to shoot is straight ahead, but the spinning propeller blocked that path. Synchronization gear, which timed each round to pass between the blades, solved the problem, with Anthony Fokker's Stangensteuerung the first system to enter service in 1915.
During World War II, piston engines grew from roughly 1000-1400 horsepower at the start to over 2000 horsepower by the end, and fighters adopted monocoque construction, laminar flow wings and swept wings. Rifle-caliber machine guns were increasingly replaced or supplemented by cannons, and German jet and rocket fighters entered combat in 1944, too late to change the outcome.
The first jet-versus-jet dogfight occurred during the Korean War on the 8th of November 1950, when an American F-80 shot down two North Korean MiG-15s. The swept-wing MiG-15 reached 1075 kilometers per hour and shocked F-80 pilots, prompting the United States to rush the North American F-86 Sabre into battle.
Recognized classes of fighter include the air superiority fighter, the fighter-bomber, the heavy fighter, the interceptor, the light fighter, the all-weather and night fighter, the reconnaissance fighter, and the strategic fighter. The fighter-bomber, reconnaissance fighter and strike fighter classes are dual-role, combining fighter qualities with another battlefield role.
Energy-maneuverability theory was developed by Colonel John Boyd and mathematician Thomas Christie, emphasizing aircraft-specific energy maintenance and fast transients over relying on high speed alone. It shaped fourth-generation fighters and, through Boyd's Fighter mafia, led to the General Dynamics F-16 Fighting Falcon, which used relaxed static stability and a fly-by-wire flight-control system.