I always find it interesting to track the history and usages of particular words. The previous poster is probably right about the word's etymology (or history), but there are sources that claim another (although very closely related) history of the word.
One process that may have been used to create the word is "clipping," a process in which one part of the word, the beginning or (in this case) the end, is chopped off. In the case of "Nazi," the final syllables "-onal" may have been clipped, turning "National" into "Nazi."
The second process that may have been used is combination of "clipping" and "blending." Some respectable sources (such as the Online Etymology Dictionary; see the link below) claim that the first two words in the name of the party -- "National Socialist" -- were shorted (or clipped) to "Na-" and "Sozi". The two shortend versions were then combined (or "blended"), according to these sources.
Either way, the word's meaning (its "semantic range") has grown considerably in the last few decades of the 20th century. The popular comedy show Seinfeld introduced us to "Soup Nazi," for example, and women activists for equal rights are sometimes still derisively called "feminazis."
The term "Nazi" was considered a slur, which no self-respecting member of the party would ever call himself or any other member. The proper name of the party was the National Socialist German worker's party, in German Nationalsozialistische Deutsche Arbeiterpartei. It commonly identified itself as the NSDAP. Its members called themselves "National Socialists." "Nazi" was the pejorative term used by Americans and British people with the intentional negative connotation. As the above answers indicate, it was taken from the first word, Nationalsozialistische, of the party name. Because of the intensely negative image of the party (which is quite appropriate) the term Nazi has been adopted almost universally as the party name.
In more recent years, a number of white supremacy organizations have arisen who call themselves such things as the Aryan Brotherhood, the Aryan Nation, World Church of the Creator, even the National Socialist Movement. All of these groups are radical racists and consider Adolf Hitler something of a hero. Although they are commonly called "neo-Nazis," they never use the term to describe themselves.
The word Nazi comes from the official name of Hitler's political party. The name in English was National Socialist German Workers Party. The word Nazi comes from the pronunciation of the initial sound of the word "national" as it is pronounced in the German language.
In modern times, the word "Nazi" has meanings that go beyond this. We of course use the word to apply to the actual members of the Nazi Party. But we also use it to refer to anyone who is a fanatic about a cause, especially one who wants to force others to agree with/be like them.
The word Nazi refers to a member belonging to Nazi part, which was a political party believing in Nazism philosophy. The political movement developed in Germany in 1920's. It started as a small organization called German Worker's Party, Hitler joined this party and quickly gained control of it in 1919. He changed its name to National Socialist German Worker's Party in 1920. The word Nazi is formed by shortening this name as pronounced in German.
The Nazi party gained power rapidly under the leadership of Hitler, following Nationalistic policies. Supported by promise of socialist reforms. Support to Nazi part increased substantially after great depression of 1929, and in elections of 1932 it emerged as the strongest party in 1932 elections. On strength of his party's position Hitler Became Chancellor of Germany in 1933 and made some quick moves to seize dictatorial power with supporter of dedicated party members. Once this happened the Nazis became the main instrument to support and perpetuate the dictatorial and oppressive rule of Hitler.