What conflict is at the center of Marx's interpretation of history in The Communist Manifesto?
Marx (and Engels) declare, in the opening statement in the section "Bourgeois and Proletarians," that the history of all past societies "is the history of class struggles." Development, evolutions, progress, and even revolutions have all been caused by class conflicts. Marx describes how societies have all been divided into classes, more generally a division between "oppressor and oppressed." The class struggles in history are also economic struggles, conflicts between those who have money and those who do not ("the haves and the have nots"). Since politics always plays a role in the distribution of money and capital (from aristocracies to state governments to global capitalism), the class conflict is inherently linked to political power as well as economic systems and any inequality therein.
In earlier epochs of history, we find almost everywhere a complicated arrangement of society into various new orders, a manifold gradation of social rank. In ancient Rome we have patricians, knights, plebeians, slaves; in the Middle Ages, feudal lords, vassals, guild-masters, journeymen, apprentices, serfs; in almost all of these classes, again, subordinate gradations.
Marx then discusses how, in his era, class conflict is essentially becoming a struggle between the bourgeoisie and the proletariat, that is to say the oppressors and the oppressed. The bourgeoisie or owning class owns the means of production (the money, capital, machinery, factories) and the proletariat (working class) own nothing other than their own bodies, which means that the only way they can make money is to literally sell themselves: sell their labor value. The bourgeoisie (capitalists) keep the profits they make and keep the proletariat at wages low enough that they (proletariat) can never rise from their position. Marx and Engels predicted that this inequality would worsen and would eventually lead to a proletariat revolution. Just as the bourgeoisie staged their own economic revolution, Marx and Engels predicted that the proletariat, having nothing to lose, would stage a more overt revolution. This is noted famously at the end of the text:
Let the ruling classes tremble at a Communistic revolution. The proletarians have nothing to lose but their chains. They have a world to win.
WORKING MEN OF ALL COUNTRIES, UNITE!
For Marx, history has been made by one class dominating the others. His prediction was that the working class, proletarians, would continue this trend of evolutions and revolutions amongst classes. However, he foresaw or hoped that the proletarians would take control of the means of production in efforts of establishing equality for all and for all to share the wealth of any profits of labor.