The major confusion in the meaning of the DotP that gets many people is the distinction between dictatorship as a political system and the Marxist concept of class dictatorship.
As an example, take the US (and to a lesser extent, the west in general):
We live in a society with a somewhat democratic form of governance, yet the bourgeoisie holds ultimate power. The existence of a democratic political system doesn't change the fact that the bourgeoisie have the disproportionate power that they do in writing our laws and selecting our lawmakers.
There is no conflict between the de facto class dictatorship of the bourgeoisie and the fact that we vote in our representatives.