The bourgeoisie is the capitalist class that owns most of society's wealth and means of production. Synonyms include "the capitalists", "the upper class", "the 1%", "the aristocracy", "those filthy rich bastards", etc.
The proletariat is the working class that owns nothing but their own labour that they have to sell to capitalists in order to survive. Synonyms include "the working class", "the 99%", "the masses", etc.
"The history of all hitherto existing society is the history of class struggles.
Freeman and slave, patrician and plebeian, lord and serf, guild-master and journeyman, in a word, oppressor and oppressed, stood in constant opposition to one another, carried on an uninterrupted, now hidden, now open fight, a fight that each time ended, either in a revolutionary reconstitution of society at large, or in the common ruin of the contending classes.
In the earlier epochs of history, we find almost everywhere a complicated arrangement of society into various 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.
The modern bourgeois society that has sprouted from the ruins of feudal society has not done away with class antagonisms. It has but established new classes, new conditions of oppression, new forms of struggle in place of the old ones.
Our epoch, the epoch of the bourgeoisie, possesses, however, this distinct feature: it has simplified class antagonisms. 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."
—Karl Marx, the Communist Manifesto (1848)
Means of production are physical, non-human inputs (facilities such as factories, machinery, some tools, infrastructural capital, and nature capital) that allow for the production of commodities and the generation of capital.
Not to be confused with the means of production, a mode of production is a system that determines what the means of production are, how they are used, and who can own them.
Some examples include: Slavery/slave-based economies, feudalism, simple commodity production, capitalism, and socialism/communism.
Private property is a legal designation for the ownership of property by non-governmental legal entities. Private property is distinguishable from public property, which is owned by a state entity; and collective property, which is owned by a group of non-governmental entities, such as workers under socialism. Private property is further distinguished from personal property, which refers to property for personal use and consumption, such as your house, your car, your video games, your food, and so on.
In short, "private property" either refers to "private ownership of the means of production" or "private productive property" (such as factories).
Although it sounds scary, the DotP is not actually a dictatorship in the same sense Julius Caesar was a dictator. Marxists believe that pretty much any society is a dictatorship by a particular class. Although we think of our current society as a democracy, in the Marxist view, it is in reality a dictatorship of the bourgeoisie, or the upper classes. This is because the capital, the land, the businesses, the means of production, and the political process all are under the control of the upper classes. The working class is effectively under a dictatorship of the upper class.
A dictatorship of the proletariat is essentially a dictatorship of the working class. So the working class would have control over the land, the businesses, the means of production, and the political process.
Thanks to this Reddit user for that answer.