Commie is derogatory slang for communist. You shouldn't call someone a commie unless you want to start a fight.
A commie is a communist, a person who favors the eventual creation of a classless society in which goods are equally shared between all citizens. This word was most common in the 1950s, during the strongly anti-communist era of the US. Being called a commie back then might result in losing a job or being ostracized by friends. Today, you're much less likely to hear the word commie, which has the same root as communism, the French communisme, from the Old French comun, "common, free, open, or public."