A form of society without government, social classes and money. The first part of that sentence is why The Soviet Union, China, Cuba, Vietnam etc... have never been communist countries. This is not to say that somehow these countries did something wrong when following Karl Marx's theory. Marxism is fatally flawed as it requires a government to gain power and then assumes that as time goes on the leaders in charge of the government will loosen the power that the government holds until it is finally free from government. It is absurd to think that the best way to destroy government is to make one, and a huge one at that.
So this is where you will make the distinction between Communism and communism. Capital C Communism is the most widely know version of the word. Communism is used when discussing Communist Parties, and state Capitalist countries like Cuba, China and the Soviet Union. Lower case c communism however is used when discussing a society without government, classes and money. Lower case c communism is also used by the libertarian sections of communist theory that advocate progressing to a communist society without creating a state that will choke all of the freedom out of the people it governs.
After understanding that you can understand that communism and Anarchism are at times the exact same thing. Most people would be shocked to find this out because most people think of state government when thinking of communism. As has been stated communism is a form of society without government, classes and money. What has been seen in the Soviet Union was an attempt at using Marx's theory on how to get to communism. Anarchism has many different strands of thought but at it's base it is a form of society without government and most hierarchical social structures(such as classes, bosses, governors, presidents, masters, slave drivers etc..). The difference is that communism is a type of society without money where as not all anarchist theories are without money.
For an example of communism/anarchism working on a large scale research The Spanish Civil War 1936-1939. George Orwell's personal experiences participating in the war(on the side of communism and against Communism and Capitalism) are written in his book "Homage to Catalonia".