Nazism or National Socialism

Yes, National Socialism (Nazism) and Communism are both forms of the catch-all term, Fascism.

Nazism has of course no connection at all with Socialism, which means a classless stateless society based on production for use.

Nazism is short for National Socialism (Nationalsozialismus in German). This was the extreme right-wing political movement led by Adolf Hitler.

The spelling for the German movement of World War II is "Naziism" (or rarely "Nazism"). The word Nazi comes from the party name Nationalsozialistische Deutsche Arbeiterpartei or National Socialists.