Fascism is a governmental structure based on a strong central government, typically a dictatorship and normally marked by strong nationalist sentiments as well as racism. National Socialist Germany and Fascist Italy are classic examples. The name fascism originated from the name of a Roman war axe known as a fasces (it constituted a magistrate's symbol of authority) that Mussolini adopted as the symbol of his party to appeal to Italian nationalist sentiment. Mussolini's famous slogan typifies Fascist thinking:
Everything in the state, nothing outside the state, nothing against the state
Obviously, National Socialist Germany is another prime example of fascism as there was a strong dictator and strong nationalist sentiment based to a large extent on racism. Although Jews are not considered a race by scientists, the National Socialists transformed them into a race. (incidentally, the term "Nazi" was pejorative: they preferred to be called "national socialists").
Communism is not based on a strong governmental structure in its purest form; in fact Marx envisioned that government would become superfluous and eventually wither away. Under true communism, all property is owned collectively (there is no private ownership) and workers work for the good of everyone in the society. It is based on worker solidarity which presumably would transcend nationalist sentiments. This constitutes a strong difference between Fascism and Communism, as the former is based on nationalist sentiments and the latter on worker solidarity. The point is illustrated by comparing the German national anthem (Deutschland uber alles) with the Communist anthem, the Internationale.
Confusion often exists in distinguishing between the two because Communism as an economic system does not work. All Communist regimes have been totalitarian in nature. Thus, although Communist in name, communist governments have strangely fascist elements.