Capitalism can exist in many forms, from those with minimal government intervention (which is libertarianism) to those in which the government plays a very large role (e.g., what exists in countries like Sweden, sometimes called "welfare state" capitalism) and those somewhere in between. In all cases, however, capitalism is a system of class inequality.
This inequality arises from the five defining features of capitalism: 1) that virtually all of the means of economic production (farm land, mines, factories, commercial-use buildings such as warehouses and office buildings and retail stores, intellectual property such as computer software, etc.) are owned as the personal property of individuals or of groups of individuals (e.g., as a corporation) but not owned by society as a whole; 2) that whoever owns the means of production, and only they, own the products or services that are produced with those means of production (typically by many people--the hired workers--who do not own the means of production*); 3) that economic products and services are commodities that are bought and sold by individuals (or groups of individuals) in the "free market" as an exchange of equal value (of the commodity) for an equal value (of money); 4) that society is based on the belief that it is both innate human nature and also morally right for people to try to increase their personal wealth as much as possible (by making a profit by buying low and selling or renting high) and to become richer than other people; the argument for this "morality" asserts that when everybody acts in their self-interest this way the result is (via Adam Smith's famous "invisible hand") that everybody in society benefits; and 5) that money is, and ought to be, power because virtually everything can be bought, not only material things but also the obedience of people (hiring an employee means buying their obedience for certain hours of the day.)
Here is how class inequality inevitably arises and increases in any capitalist society. Even if one imagines a capitalist society starting out with everybody equal in terms of their personal wealth, inequality in wealth will inevitably emerge merely because of facts such as that some people are lucky and others unlucky (one person's house burns down and another's doesn't, one person has a crop failure and another doesn't) and some people have many siblings and inherit very little while others are an only-child and inherit more, etc. etc., as discussed here and here. This wealth inequality means that the society will eventually have three kinds of people: #1) Capitalists: those who own means of production on a scale so large that the productive use of them requires the labor of those who do not own them. #2) Workers: those who do not own sufficient means of production (or, typically, any means of production) with which to provide for themselves (directly or by trade) enough to live satisfactorily and who therefore either sell their labor to capitalists for wages in order to obtain money to buy what they need to live, or rely on public welfare or private charity. (Government employees are workers whose employer is the capitalist class as a whole, which uses the government for its collective needs.) #3) Small business owners: those who own sufficient means of production with which to provide for themselves (directly or by trade) enough to live by using only their own (or family members') labor, but who do not own enough means of production that would require the labor of many others to be productive. (Self-employed professionals such as doctors and lawyers are a kind of small business owner. Their "means of production" is their education and office and books, etc., and their "product" is the service for which their clients pay them.)
(Defenders of capitalism don't like to talk about how capitalists actually came to own so many of the means of production in the first place, preferring to say "They bought them fair and square." But if one asks, "How did the person they bought them from come to own them?" and work backwards in history this way, it usually turns out that the first private owner of the means of production seized them violently from people who used them but did not consider them to be anybody's private property. Land was seized violently from the tillers of the land by early feudal lords. European upper class people seized land from the natives in the New World and in other colonies in Africa and Asia, and forced "not-so-lucky" people (indentured servants, slaves, prisoners, etc.) to work the land. Capitalism is based on ruthless violence; it is not "natural.")
Capitalists grow richer than workers because in any capitalist business only the capitalists own the commodity produced by the workers; hence only the capitalists possess the money obtained from selling the commodity, and only the capitalists have a say in determining how much, if any, of the profit (the sale price of the commodities minus their cost of production) goes to the workers and how much to the capitalists. The "admirable" (in the morality of capitalism) reason the capitalists go into business in the first place is to get richer, so it is not likely that they would take for themselves less of the profits than possible by, say, sharing their profits with their workers or, say, freely sharing their commodities with others according to need rather than selling them for their full value to obtain the maximum profit.
Small business owners cannot grow rich the way capitalists can because they can only produce a very limited amount of commodities (in a retail store the commodity is the service of making a product conveniently available to the customer) with only their own labor, and hence can obtain only a very limited amount of profit from their sale, compared with a capitalist who can profit from the sale of an enormous amount of commodities produced by the labor of many other workers.
Not only do capitalists grow richer than workers, they also are, in a very real sense, masters in a master versus "wage-slave" relationship with workers. When a worker accepts a job from a capitalist he or she must agree to obey the "on-the-job" commands of the capitalist or else be fired. Within the capitalist business there is not even a pretense of democracy--it is an overt dictatorship of the owners over the workers.
The result of the above is that capitalists gain greater power in society than workers because they grow richer and because money in a capitalist society is power, and capitalists also have direct power over workers "on the job." Based on these sources of unequal power and privilege, capitalists form a dominating upper class over the working class. Capitalists use their greater amount of money than workers have to achieve yet more power and then use that power to obtain yet more money (for example, by controlling the government and using it to enrich themselves in countless ways, and by pitting working class people against each other to prevent them from being united in solidarity against the capitalist class.) A capitalist society is thus a society in which there is always a class war between the capitalists trying to maintain or even increase their unequal (compared to workers) wealth and power and privilege versus workers trying to make society more equal and democratic and based on people helping each other.
To prevent workers from making society more equal and democratic, the capitalist class needs to use a combination of a) cowing workers into submission, b) pitting workers against each other so as to divide-and-rule them, and c) making workers believe that they are, in some way, inferior human beings who do not deserve to live as well and have as much say in society as capitalists (one way of doing this is by treating working class people like dirt, as discussed here). Many of the worst aspects of our present society stem from capitalists doing these kinds of things, such as 1) using standardized "high stakes" tests in public schools that have--by design--a built-in failure rate so that no matter how well children learn their lessons the failure rate will be what the test-makers want it to be, the purpose of which is to make working class children believe they are not smart enough, or didn't study hard enough, to deserve a decent-paying job; 2) waging Orwellian wars of social control based on lies (e.g., Saddam's "Weapons of Mass Destruction") in order to make ordinary people believe that it is unpatriotic not to obey their capitalist rulers (i.e., the politicians who are beholden to the capitalist class), 3) making workers of different races mistrust and fear one another and 4) forcing workers to work for lower wages by telling them that if they don't the company will hire other workers (often foreigners) who will work for those lower wages. Click here to see how many if not most of the worst problems we face today are caused by capitalism and solved by egalitarianism.
* The problem of class inequality remains even if the workers in a capitalist economic enterprise are the only owners of it, as discussed here.