Class is a structural concept that picks out the role that an individual occupies in the economic structure of society. A person's class specifies their relationship to the means of production in their society. Class relations are, accordingly, relations of production. They describe where an individual fits into the process whereby a society produces the necessities of life. A class that owns and controls the means of production is in a dominant position vis-a-vis other non-owning classes and is thus a ruling class. The means of production (which includes all of the technology, factory equipment, tools, technically exploitable knowledge, raw materials, etc. needed to produce socially-necessary goods) is a dominant good because the survival of everyone in a society depends on it. This places capitalists, who exclusively own and control the means of production, in a position of considerable power.
Accordingly, in capitalist societies non-owning individuals depend on the capitalist class in two primary ways. First, the society's means of production are needed to produce all the goods (food, clothing, shelter, and so on) that are needed to subsist. So if individuals are to acquire most of the goods they need to survive, they have to purchase them from capitalists. If capitalists aren't selling, or if they're selling at prohibitive prices, these individuals cannot acquire the basic necessities of life. Second, individuals who possess no means of production are dependent on capitalists for employment. In order to earn an income, these individuals must sell their ability to work to those who own the means of production, i.e. to capitalists. If capitalists aren't hiring, these individuals find no work and hence earn no income.
This is the foundation of the social power of the capitalist class. If we didn't need them to acquire the necessities of life or employment, their social power would largely dissolve. They of course have other sources of power (e.g. the state is structurally dependent on capitalist profits in order to fund its activities via taxation, politicians are often drawn from the ranks of the ruling class themselves or are financially connected to the class, if politicians aren't owned outright by capitalists the basic media organizations are, etc.). But their basic power lies in the fact that they own and control something we all need to survive- the means of production.
Notice that I've said nothing about income inequality, "socio economic status", opportunity hoarding, class identity, or the like. What we've been talking about is relations of domination- social relations in which there is an asymmetrical distribution of power. Once we have a view about how power is diffused through these social relations, only then are we in a position to understand how various disparities (in income, or opportunities, say) come about. It also stands to reason that conception of class would shed some light on how identities and self-understandings form among the members of certain classes, how they are able to procure or hoard certain opportunities, and so forth. Of course, a full analysis of these matters would have to also take into account certain sorts of oppression (e.g. racial oppression) that some, and not others, suffer. But let us just focus on class for the moment. Take the example of income.
If we want to understand income inequality, we shouldn't start from premises about an individual's intelligence, their productivity, their individual ambition, or their deservingness. We look at where they are situated in the economic structure of society; income levels mostly reflect institutionalized roles/offices rather than individual productivity or effort. If you own and control sizable amounts of productive capital, you are in a position of social power and, hence, are able to acquire a rather large portion of the overall social product. As C.B. MacPherson puts it, ownership and control of productive capital equips one with extractive power over non-owners. This is the power to exploit the labor of others and thereby extract value from their efforts.
Those who own no land or capital (no means of production) have no extractive power. As MacPherson puts it, "in a full capitalist society, with its substantial concentration of ownership of capital and productive land, a few men have extractive power over many; hence each of the few has extractive power equivalent to the whole power of several other men. The greater the concentration of ownership of capital, the greater the proportion of each owner's entire power consists in his extractive power."
So class isn't in the first instance about different levels of consumption among individuals. Nor is it primarily about the distribution of wealth, which (as is often noted), is far more unequally distributed than income. Nor is class, at rock bottom, about one person having expensive, fancy clothes and another lacking such "nice things". Class doesn't track whether isolated individuals are off in their respective corners consuming unequal amounts of stuff. Class tracks social relations between individuals by examining the diffusion of power through the economic structure of society. Class analysis aims to show us how relations of power enable some individuals to extract benefits from others, thereby amassing large amounts of wealth and so forth. Wealth represents the spoils, not the primary source of domination.
What this makes clear is that the ire of left-liberals (i.e. income and wealth inequality) is not sui generis. It is the result of a oppressive social relations built into the economic structure of society. It is, in short, the result of capitalism. As Marx wrote in the Critique of the Gotha Program:
Any distribution whatever of the means of consumption is only a consequence of the distribution of the conditions of production themselves. The latter distribution, however, is a feature of the mode of production itself.Similar analyses would be required for gender domination and racial oppression as well. Neither are directly caused by income or wealth inequality (though they are both reinforced and consolidated by such inequality). Causation mainly goes the other way around: racial oppression is what explains the wealth gap between, say, white and black people in the US. And of course, we shall not be able to explain or understand this racial disparity without the concept of class, for it is not the case that the resulting distribution of wealth and income is uniform throughout the black population (just as it is not the case that the distribution of black people throughout the class structure is even or uniform).
What this suggests is that the concrete functioning of racial oppression requires a structural, not merely individual experience or identity-based, analysis. Given that racism is inscribed in the basic structure of social institutions, and given that we live in a capitalist society, it follows that racial oppression and class are inextricably bound up with one another. Racial oppression must therefore be given a materialist analysis, i.e. an analysis and critique that focuses on the material conditions of social life. The critique of cultural racism, micro-oppressions, whiteness, and so forth are all crucial to understanding and uprooting racist oppression. But these critical enterprises must, at the end of the day, be related back to the material conditions of social life if we are ever to have any viable political practices for collectively challenging the institutional and structural oppression. We don't need more "tolerance" or "recognition" from dominant groups, we need movements from below that can successfully challenge the social relations of domination themselves.