Founded during World War I with the emergence of military aircraft, it was dissolved after the war, as a part of the Treaty of Versailles. On February 26, 1935, Adolf Hitler ordered Hermann Göring to re-instate the Luftwaffe, although the treaty was still in power.
By 1939, on the eve of the outbreak of World War II, the Luftwaffe had become the most powerful air force in the world. As such it played a major role in Germany's early successes in the war, and formed a key part of the Blitzkrieg concept, much thanks to the use of the innovative Junkers Ju-87 dive bomber plane (Sturzkampfflugzeug -- "Stuka"). A contingent from the Luftwaffe was sent to support the rebel side in the Spanish Civil War with planes (notably the Ju-87) and personnel.
The inability of the Luftwaffe to control the skies in the Battle of Britain after the tactical mistake of bombing industrial targets in cities instead of British airfields formed a key point in the war, and German air power diminished further with the arrival of the Americans, though it remained strong, especially on the Soviet front. The Luftwaffe were the first air force to fly jet fighters, with for example the Heinkel He-178 (1939), the Messerschmitt Me 163 (1941) and the Messerschmitt Me 262 (1942).
(more needed on post-war history)