The word "capitalism" hides a multitude of meanings. Marx and Engels, whose writings spawned the eponymous ideology of Marxism, while they didn't coin the term "capitalist" which they used in The Communist Manifesto (1848) and Das Kapital (the first person we think of now as an economist to do so in the English language was David Ricardo, in his "Principles of Political Economy and Taxation" (1817)), probably did more than anyone else to popularize the notion that the prevailing economic order of their day was a capitalist one. Max Weber's book "The Protestant Ethic and the Spirit of Capitalism" (1904), in turn, is perhaps the defining moment in which apologists of the status quo that Marx critiqued took ownership of and embraced the label that he had given them.
In the Marxist sense, the essence of a capitalist economic system is that it is one in which what modern slang would call the One Percenters a.ka. the Bourgeois a.k.a. the Capitalists a.ka. the wealthy, run the economy, with the assitance of the Petite Bourgeois, a.k.a. the Upper Middle Class of managers, professionals and small business people, while subjecting everyone else, the Proletariat a.k.a. the working class, to wage slavery and/or poverty in a way exploiting them to generate the riches the Capitalists enjoy.
Of course, self-described supporters of capitalism don't generally see the oligarchy of the wealthy as the really essential component of the the economic systems that they support.
Instead, capitalists generally focus on the notion of a decentralized market based private sector economy, as opposed to one run by or owned by the state. The term "laissez-faire" emphasizes a lack of government regulation of eocnomic transactions, as does the term "free market."
A common way to operationalize a distinction between capitalism and communism on a continuum is to look at the percentage of the economy (for example, as a percentage of GDP or employment) that is in the governmental as opposed to the non-governmental sector. At one extreme is the fully market based economy in which everything is handled through a bare minimum of governmental institutions to the point of a libertarian or anarchist extreme, and at the other is a "command economy" in which everything is owned by the government and all economic decision making is made on the basis of central planning (like the military). In between are way stations like regulated capitalism, a "mixed economy", and democratic socialism.
Some of the varying definitions of capitalism sound good and well, until one starts trying to apply them to specific facts and finds that they start reaching different conclusions.
A Madisonian society in which Yeoman farmers with roughly equal holdings interact with each other and small business people, govern themselves via small, democratic political units, and cede power to centralized economic and political institutions only when absolutely necessary in a manner that retrains profits and control of the small enterprises at the bottom of the heap, could very well be a "free market" system without being "capitalist" in the Marxist sense of being dominated by wealthy members of a capitalist class.
In truth, many Americans who think of themselves as capitalists are just as suspicious of the concentration of economic power in the hands of a wealthy few as Marx was himself. While the Communist economic ideology built on a foundation of Marxist thought certainly didn't work well, or end well, if you read "The Communist Manifesto" in the frame that its original readers would have received it, without the historical gloss, it isn't hard to imagine its predictions about the inevitable demise of the capitalist order, and its observations of the ills of system in which wealth determines who makes decisions, producing an approach to addressing these ills that could have been far less bloody and far more workable in practice.
Indeed, one of the classic steps in the economic transition of a country from having a Third World economy, to a democratic capitalist economy, is to break the economic dominance of a narrow class of the superrich. It is notable in this regard that the political leaders of supposedly Communist China have far more personal wealth than the political leaders of supposedly the Capitalist United States. In the American South, the demise of the Plantation system of farming that the Civil War and Emanicipation provided a final death knell for, coincided almost immediately in the antebellum era with economic development and technological moderization for a region that had long lagged behind the rest of the nation.
When you look at examples, it pretty soon becomes clear that the market place choice emphasized by Milton Friedman is really more central to what makes those who embrace the idea of "Capitalism" (e.g. in his book "Capitalism and Freedom") than the distinction between the governmental and non-governmental sector.
Consider a couple of examples from contemporary Denver.
Denver has a system of "charter schools" (and several different flavored variants on the concept like magnet schools or magnet programs) which are autonomous public schools that students may choose in a structured "school choice" system, give rise to something much closer to the ideal of how firms ought to behave in a capitalist market economy, despite being formally within the governmental sector. Charter schools don't compete based on price, but there is a vigorous competition in the school choice system between each and every school, charter or otherwise, involved in the school choice process that has real world consequences for the competitors. Schools that are empty because nobody chooses them eventually get shut down, regardless of their formal character as a charter or magnet or neighborhood school. Improved school choice options and this competition have, in turn, dramatically increased the percentage of school aged children in Denver that choose to enroll in schools under the Denver Public Schools system school choice umbrella, while alternatives like Roman Catholic parochial schools in the private sector have seen their market share contract.
In contrast, while the emissions testing program in Colorado is operated by a private sector government contractor, this monopolist operation shows all of the characteristic pathologies we associate with government owned monopolies of particular types of economic activities, without any of the redeeming virtues of government bureaucracies: they don't even have ATM machines or take credit cards (unlike many state agencies in Colorado) and should you forget to bring cash or a checkbook, they are perfectly happy to futilely run a test twice on the same car in one day even though nothing in their contract prevents them from printing out test results the first time and then holding the results hostage until payment is made (since everybody is getting tested to get a piece of paperwork that establishes compliance with environmental laws, rather than to actually get the information that the testing provides), like any ordinary enterprise in a competitive situation where customer satisfaction matters would have done. They also have the long lines and indifference to customer wait times that are characteristic of governmental monopolies, since both have captive audiences of mandatory customers and no competition.
Freed of ideological baggage, people are often quite ardent in their support of government economic planning, for example, in the form of zoning codes that regulate real estate development under the watchful eye of a Planning Commission, and rather less supportive of a "lassiez-faire", free market approach to real estate development, even though many of the most loved features of urban land use emerged in the absence of meaningful zoning and land use regulation.
Likewise, some of the most die hard rhetorical opponents of command economies and government ownership of economic resources on the political scene think that the military's subculture is a real world utopia, aren't alarmed that top civilian and military executives are paid far less than comparable private sector executives, and can't imagine anything amiss in a monopsomy system of defense procurement that permits bids only from domestic companies in any case where there is any domestic company that can meet the military's needs, while considering government owned military manufacturing enterprises to be unthinkable.
A sizeable part of the grass roots Republican base consists of moralist big government conservatives with a populist bent, who have no real objections to taking 21st century capitalists, like GOP Presidential candidate Mitt Romney, down a peg or two. It wasn't President Obama who called him, in a classic Marxist paradigm, a "vulture capitalist." In 2008, Mike Huckabee was also something of a big government populist candidate within the GOP, and Mitt Romney's closest rival, Rick Santorum, during his service in the U.S. Senate and on the campaign trail, has not been a friend of international free trade. Santorum's views on international trade are basically merchantalist, the very economic doctrine that Adam Smith, the patron saint and founding father of modern capitalist economic thought, put at the heart of his own critique of the economic status quo.
Does it matter that the essence of what we think of the essential elements of a modern, functional Western style economic system don't include a dominant role for people whose primary qualification is vast wealth? Should we care about the nuances of the terms we use now that they have acquired secondary meanings of their own that almost ignore the original meaning of Ricardo and Marx?
I think that it does. A word like capitalism almost inevitably has buried within it acceptance of the inherent natural supremacy of the titans of industry without whom the economy would shrivel assumed by Ayn Rand in "Atlas Shrugged" that there is little reason to believe is an empirically valid thought experiment in a context like the modern American economy, even if that notion may have been pretty valid in contexts like Zimbabwe, as illustrated by the catastrophe that land reform there created. Capitalists may be necessary in Third World economies where the number of people with the skill sets needed to operate the highly productive economic and reasonably just political institutions of a modern Western style political economy are in scarce supply. But, it is far less obvious that capitalists are really necessary in well educated, mature political economies where there are far more people competent to handle skilled managerial and technical responsibilities than there are opportunities that fully utilize those talents.
So long as we use "capitalism" to describe our economy, we are inherently legitimatizing the "divine right" of plutocrats to rule, rather than forcing them to justify their power in a utilitarian way, i.e. to earn it.