A very interesting subject, my personal definition for each is:

Communism - An economic philosophy based on common ownership of all lands and businesses. Private ownership is restricted to personal items. Historically, communism is usually accompanied by a Fascist government.

The word communism and the philosophy of common ownership (commune) has been around for over 500 years.

Fascism - A form of government dating back to the Roman Empire. It is essentially a concentration of power in a single leader (dictator).

As construed by Mussolini in 1920 it was anti-communist but it was considered by many as a modern, 20th century answer to Marxism.

Hitler's fascism originated as a revolutionary socialist system but it evolved to heavily managed capitalism as Hitler curried favor with the German upper class.