People's social and political beliefs are shaped by economic relations in general, and shaped by how each individual participates in those economic relations. And people tend to group themselves socially and politically according to their commonalities in how they participate, i.e. their economic class. There are, of course, additional factions, sub-groupings and cross-groupings, but economic class solidarity and commonality is an important concept. The capitalist ruling class is especially aware of and adheres to a concept of class solidarity, solidarity that survives even very deep factional differences.
We can identify three primary classes in modern society. First of course is the capitalist class, the class of people whose primary economic activity consists of owning the means of production and renting that means of production to others to perform useful work. They might own physical capital, i.e. factories and machines; more often they own financial capital, i.e. the money used to purchase labor to effect actual production. Opposite the capitalist class are those who sell their labor power to the capitalists and who actually do the production. Members of the capitalist class often do actually perform labor: they typically have to administer and manage the use of their capital. The point is not that they don't work, but rather that they don't sell their labor to someone else.
Within this second group are two classes: the professional-managerial class and the working class. The professional-managerial middle class typically possesses "human capital", i.e. education; more specifically credentialed education, where there is some limit in theory and practice on the availability of the credential. Physicians and lawyers, who have to have a MD or JD, are the most clear-cut examples of credentialed members of the professional-managerial middle class. The working class are those who sell their labor power to the capitalists without relying on some limited-availability credential.
The class structure of society is not a set of hard-and-fast divisions; it is, rather, a loose and abstract way of looking at general commonalities in attitudes and circumstances. Modern economic relations are extremely complicated, and many individuals can operate as if they were members of each of these classes in the course of a day. There are fine gradations within each class: both radiologists and physicians rely on credentials, but the value of those credentials are very different. There are also additional classes outside the working/professional/capitalist system, such as students, criminals, the unemployed, the hyper-exploited working class, etc.. But there are real and important generalities we can draw about the social and political behavior of people by looking at their economic class.
One of the interesting features of the capitalist system is the distinction between the formal government and the capitalist ruling class. Before the capitalist republics, the ruling class was the highest level of government. In contrast, members of the government — even elected members or those appointed directly by elected members — are often not members of the capitalist ruling class. Presidents Obama and Clinton, for example, were members of the professional-managerial class, albeit in the highest level of that class. On the other hand, George W. Bush was himself a member of the capitalist class. And every government since ancient Sumer and Egypt has relied on an army of professional-managerial bureaucrats, who have formed a distinct economic class.
The important observation about class and government, though, is that elections notwithstanding, the formal government is more-or-less* dominated by the capitalist class. And not just the government: the domination of the capitalist class pervades all of our social, political and economic relations. One obvious example is the aversion of members of the working and professional classes to discuss their salaries, which is a direct denial of the "perfect information" component of a free market. This aversion is a social construction within members of the working class that furthers no interests but those of the capitalist class.
*In addition to the class struggle of the working class, there's also a class struggle between the professional-managerial class and the capitalist class. I suspect this struggle explains a lot of the 20th century, especially the American "golden age" immediately following the Second Imperialist War, where the professional-managerial class achieved a degree of ascendancy over the capitalist class, due to the class "treason" of Franklin Roosevelt, himself a member of the capitalist class.
Our government as well as our pervasive social and political constructions operate for the interests of the capitalist class and against the interests of the other classes, especially the working class. Any time these constructions operate against the interests of a class, they are by definition oppressive and exploitative. This domination therefore acquires the label of dictatorship, i.e. the dictatorship of the bourgeoisie. The immediate goal of communism transitioning from capitalism then is to create a formal government and pervasive social and political constructions that operate for the interests of the working class and against the interests of the capitalist class (and, to a certain extent against the interests of the professional-managerial class), i.e. the dictatorship of the proletariat. The capitalist class is not going to disappear overnight, even after their resistance to losing their status as the ruling class has been successfully overcome. If they are not to regain power and restore the capitalist system, their interests must to some degree be actively worked against, i.e. they must be oppressed.
There's a big difference between the the domination by the capitalist class and the proposed domination by the working class: the oppression and exploitation of the working class is interminable: the capitalist class must have a working class, which must be interminably oppressed to maintain the position of the capitalist class as the ruling class. The working class, however, does not need the capitalist class; the oppression of the capitalist class is terminable. The capitalist class cannot absorb the working class, but the working class can absorb the members of the capitalist class and eliminate the class. (I'm talking about eliminating a class, not the people in it. We don't need to kill the members of the capitalist class to eliminate the class, we merely need to invite the capitalists into the new ruling class by making them workers.)
My ex-wife2 gave me (perhaps inadvertently) an interesting analogy. There are pervasive social and political constructions (and, until very recently, many formal government and legal measures) that operate for the interests of men and against the interests of women. Collectively, these measures deserve the label of the "patriarchy". Resistance to and destruction of these measures commonly goes by the label feminism. While the overall goal of feminism is for the most part to establish equality between the sexes, labeling this movement as something like the "sexual equality" movement fails to rhetorically capture the primary intent to actively destroy the patriarchy, to extirpate male privilege. In an attenuated metaphorical sense, this is the cake-splitting problem: the patriarchy demands the whole cake; to demand only half the cake encourages a compromise where men get three-quarters. Demand (perhaps apparently "unreasonably") the whole cake, and the compromise of equality becomes more tenable. Obviously the analogy is not exact: men cannot be absorbed into women in the same sense that the capitalist class can be absorbed into the working class. Still, the analogy illustrates the important concept that the dictatorship of the proletariat is not just about establishing democracy, it is also about actively extirpating the power and privilege of the capitalist class so we can have an effective democracy. (Again, I'm talking about extirpating ideas and social constructions, not killing people, in the same sense that we can extirpate the idea of male privilege without having to actually kill every man.) As an exercise, think about the analogy between the dictatorship of the proletariat and the idea of affirmative action to correct historical racial discrimination.
Dictatorship is a harsh word, and invites the equivocation between the dictatorship of the proletariat and an actual dictatorship for (or supposedly for) the proletariat. Sadly, even actual communist regimes have justified actual dictatorships supposedly for the proletariat, which have unsurprisingly degenerated into dictatorships for the dictators and the governmental class. This objection has a lot of persuasive power, and argues for alternative constructions such as the "proletarian state" (a term with some currency in communist theory). On the other hand, we have an obligation to be honest and upfront. Because the capitalist class justly sees their maintenance as the ruling class to be in their interests, they justly see their removal as the ruling class to be against their interests and constitutes oppression. The only honest communist answer is, "Yes, you're absolutely correct. We aim to actually oppress you. We can live with that." If we really do aim to oppress a class of people, we should come right out and say so, and adopt a label that makes that oppression obvious. There's no doubt that the dictatorship of the proletariat does make that position clear.