From Wikipedia:

The full name of the party was Nationalsozialistische Deutsche Arbeiterpartei (English: National Socialist German Workers’ Party) and they officially used the acronym NSDAP. The term “nazi” had been in use, before the rise of the NSDAP, as a colloquial and derogatory word for a backwards farmer or peasant. It characterised an awkward and clumsy person, a yokel. In this sense, the word Nazi was a hypocorism of the German male name Igna(t)z (itself a variation of the name Ignatius)—Igna(t)z being a common name at the time in Bavaria, the area from which the NSDAP emerged.[17][18]

In the 1920s, political opponents of the NSDAP in the German labour movement seized on this. Using the earlier abbreviated term “Sozi” for Sozialist (English: Socialist) as an example,[19] they shortened the NSDAP’s name, Nationalsozialistische, to the dismissive “Nazi”, in order to associate them with the derogatory use of the aforementioned term.[20][18][21][22][23][24] The first use of the term “Nazi” by the National Socialists occurred in 1926 in a publication by Joseph Goebbels called Der Nazi-Sozi [“The Nazi-Sozi”]. In Goebbels’ pamphlet, the word “Nazi” only appears when linked with the word “Sozi” as an abbreviation of “National Socialism”.[25]

After the NSDAP’s rise to power in the 1930s, the use of the term “Nazi” by itself or in terms such as “Nazi Germany”, “Nazi regime”, and so on was popularised by German exiles outside the country, but not in Germany. From them, the term spread into other languages and it was eventually brought back into Germany after World War II.[21] The NSDAP briefly adopted the designation “Nazi” in an attempt to reappropriate the term, but it soon gave up this effort and generally avoided using the term while it was in power.[21][22] In each case, the authors typically referred to themselves as “National Socialists” and their movement as “National Socialism”, but never as “Nazis”. A compendium of Hitler’s conversations from 1941 through 1944 entitled Hitler’s Table Talk does not contain the word “Nazi” either.[26] In speeches by Hermann Göring, he never uses the term “Nazi”.[27] Hitler Youth leader Melita Maschmann wrote a book about her experience entitled Account Rendered.[28] She did not refer to herself as a “Nazi”, even though she was writing well after World War II. In 1933, 581 members of the National Socialist Party answered interview questions put to them by Professor Theodore Abel from Columbia University. They similarly did not refer to themselves as “Nazis”.[29]