There are several different trends of communists. The only real communist movement nowadays is the 'Marxian' one, started by Marx and Engels. Out of the 'Marxians' there are mainly Marxist-Leninists and Maoists. Marxists-Leninists include Marx, Engels, Lenin, Trotsky, Guevara and Castro (althrough the last two were influenced by Maoism a little). Marxist-Leninism was the main trend in early Soviet Russia and currently in Cuba.

China and Vietnam were the main examples of Maoism and currently there is an Indian and Nepalese Maoist rebellion.

There are differences in these trends, obviously, but we all want one end goal: a society that has no classes, no state, no army and no money. Factories etc are run by the people who work in them and the community distributes goods according to need. Because only what is needed is produced, people can spend their time doing whatever the hell they want to. We call such a society communism (lower case 'c').

To get to there society needs to pass through a transition period called socialism or Communism with a capital 'C'. In this period, there is a state and there are classes and money etc etc. But the working class are the dominant class, and they prepare society for communism.

This is really simplistic and brief. There have been hundreds of Communists who dedicated their entire lives to contributing to the cause. Below are a couple of works to get you started:

Principles of Communism (Engels, 1847)
Communist Manifesto (Marx, 184. (Only bother with chapters 2 & 3).

Find them at marxists.org