The bourgeoisie are the rich, the owners of the means of production (factories, agricultural land, industrial machinery) and the owners of capital (profit and financial products like stocks and futures). Unlike the proletariat or lumpenproletariat, they have spare change lying around that they can use to make more money by hiring people, investing, and so on.
The Manifesto calls the bourgeoisie "the exploiting and ruling class" (Preface.6). These capitalists are the exploiting class in that they take—or steal, Marx might say—profit from the proletariat's work. They're also the people in charge, according to Marx, since governments are merely committees for managing the bourgeoisie's affairs (Section1.12).
In other words, according to Marx, the government functions in the interest of the rich, whether that's the very rich or even just those who wish to join their ranks. While democracies claim to represent the interests of the people as a whole, Marx would say it's unsurprising that the United States is actually controlled by few on top.
Marx spends Section 1 of the Manifesto chronicling the rise of the bourgeoisie over the aristocracy, going so far as to refer to the capitalists as having "played a most revolutionary part" in history (Section1.13). In fact, throughout the 1848 revolutions and later, Marx and Engels often supported the bourgeoisie when it acted against the feudal aristocracy (Section4.5). They tended to believe that absolute monarchs had to be cleared out of the way by the capitalists before the proletarian show could take the stage.
The Discreet Charm of the Bourgeoisie
Yet Marx didn't like the bourgeoisie—at all. According to the Manifesto, the bourgeoisie, "wherever it has got the upper hand, has put an end to all feudal, patriarchal, idyllic relations. It has [...] left remaining no other nexus between man and man than naked self-interest, than callous 'cash payment.' [...] It has resolved personal worth into exchange value [price], and [...] has set up that single, unconscionable freedom—Free Trade. In one word, for exploitation [by the aristocracy], veiled by religious and political illusions, it has substituted naked, shameless, direct, brutal exploitation" (Section1.14).
In other words, when the capitalists run the show, everything boils down to self-interest and money. For example, if someone asks how much someone else is worth, the question is now generally taken as asking how much money that person has. Some supporters of capitalism, such as Ayn Rand, see that as a good thing. They'd argue that it's virtue that drives money-making, and that if some people are unemployed and not rich, they should blame themselves.
Since the "Great Recession" financial crash of 2008 and the Occupy Wall Street movement, it's become common to speak of the bourgeoisie as the 1%—the people who are in the top 1% of income-earners—and oppose them to the 99%, or everyone else. This is a called a Manichaean outlook: it divides the world into two opposing forces, one bad, one good. Marx certainly tended toward Manichaeism: "Society as a whole is more and more splitting up into two great hostile camps, into two great classes directly facing each other: Bourgeoisie and Proletariat" (Section1.5).
Now, such a crude lens on the world, when combined with Karl's push for revolution, is reminiscent of the communist Khmer Rouge, who wanted to eliminate "bad" intellectuals in favor of the "good" peasantry, and who "solved" the problem by shooting people who wore glasses. No glasses? Good person. Glasses? Bad person.
Who's a Bourgeois?
Marx's division of people by class can spark what philosophers call a boundary problem: where do we draw lines as to who's in a particular class and who's not? Individual billionaires have the largest say over the means of production, but what about their children? They might not yet have such ownership, so are they or are they not members of the bourgeoisie? Where do we classify the petit bourgeoisie (such as small business owners), the members of bourgeoisie who defect to the proletariat (such as Marx and Engels themselves), or others whose position is not as obvious as that of, say, Bill Gates?
Unclear boundaries aside, the saying "the rich are different" exists for a reason: a person's class of origin generally leads to a life with vast advantages over the rest of the population. Consider the immunity from prosecution for killing granted on the grounds that a sane teenaged defendant's multi-millionaire parents didn't raise him with enough sense of responsibility—a condition that has been called affluenza, a mash-up of the words affluent and influenza. Is Marx right that there's something wrong at the heart of the bourgeoisie?
The Bourgeoisie's Timeline