1930, noun and adjective, from German Nazi, abbreviation of German pronunciation of Nationalsozialist (based on earlier German sozi, popular abbreviation of "socialist"), from Nationalsozialistische Deutsche Arbeiterpartei "National Socialist German Workers' Party," led by Hitler from 1920.
The 24th edition of Etymologisches Wörterbuch der deutschen Sprache (2002) says the word Nazi was favored in southern Germany (supposedly from c.1924) among opponents of National Socialism because the nickname Nazi, Naczi (from the masc. proper name Ignatz, German form of Ignatius) was used colloquially to mean "a foolish person, clumsy or awkward person." Ignatz was a popular name in Catholic Austria, and according to one source in World War I Nazi was a generic name in the German Empire for the soldiers of Austria-Hungary.
An older use of Nazi for national-sozial is attested in German from 1903, but EWdS does not think it contributed to the word as applied to Hitler and his followers. The NSDAP for a time attempted to adopt the Nazi designation as what the Germans call a "despite-word," but they gave this up, and the NSDAP is said to have generally avoided the term. Before 1930, party members had been called in English National Socialists, which dates from 1923. The use of Nazi Germany, Nazi regime, etc., was popularized by German exiles abroad. From them, it spread into other languages, and eventually was brought back to Germany, after the war. In the USSR, the terms national socialist and Nazi were said to have been forbidden after 1932, presumably to avoid any taint to the good word socialist. Soviet literature refers to fascists.