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Communists believe that in between the overthrow of capitalism and the realization of Communism (stateless, classless society) there is a transitory phase in the middle called the "dictatorship of the proletariat".
In Marxism all class societies are referred to as "dictatorships" of one class. Capitalist societies are called a "dictatorship of the bourgeoisie". The difference, and this is why Marxist countries tend to be one-party states, is that bourgeois societies have to obscure their true nature by pretending that the state under capitalism is democratic and represents the interests of all of the citizens, "the will of the majority" and all that.
Marxists consider bourgeois democracy to be a farce, and only democratic in name; and that even though there are multiple political parties with differing views, almost all of them are compatible with capitalism and represent different factions of the bourgeoisie and the petty bourgeoisie, and even labor, socialist, and communist parties have been co-opted to a certain degree by capitalism and have lost their working class nature and revolutionary potential.
Under the dictatorship of the proletariat, however, Marxists see no need to obscure the true nature of the state. Only the proletariat and its allies (the peasantry and some other minor class-strata) are allowed political representation, and Marxism is seen as the objective truth.
All societies have strict parameters, an "Overton Window" of acceptable ideological variation. Ideological discourse is allowed within these parameters, but not outside.
The Communist Party must not be thought of as a political party in the same sense as those in multi-party systems, like the Democrats or Republicans, Libertarians or Greens. Essentially, the Communist Party is the government and having one party is largely the same as having no parties.
Individual candidates are voted on due to their policies and their character, not because of a letter next to their name.