In a perfect world, a communist regime would control the government, and all "private sector" entities we know of in a capitalist society.
Government controls all means of production, wealth, and "private" property. I say private in quotations because in a communist society, there is no private property. The system is deigned to redistribute wealth and power to the poor and middle class. Means of production are highly regulated by the government and only produce what is necessary for society. No individual could own his/her company and produce their own goods, since everything is owned by the state.
On paper, it sounds fantastic because everyone gets paid equally, but in reality, the system has been tried and failed in MANY nations.