Really, Nazism and fascism are essentially the same thing. Nazism is just a form of fascism with specific ideas (like anti-Semitism) added on to basic fascist ideas.
Fascism and communism are generally said to be polar opposites on the right-left ideological continuum. However, they really have a great deal in common. In both, the state has totalitarian power over its people. The people are expected to behave in ways and believe in things that the state chooses. Neither of these forms of government wants to have anything to do with democracy or individual rights.
The main difference between the two is in their attitude towards property and, to some extent, class. Fascism does not do away with private property. Instead, the state tends to work along with the owners of the major industries in a way that will benefit both. By contrast, in communism, the state is explicitly opposed to private property and to the classes that own a great deal of it. The state wields its power in an attempt to destroy the upper class and their property, not to work with it.