Fredrick Engels writes in the "Principles of Communism" that,"Communism is the doctrine of the conditions of the liberation of the proletariat."(1)
So this then begs the question,"Just what is the proletariat?"
Well a proletariat is a class of people, so the answer lies here. In Marxist analysis of capitalist society,"Class" is defined by your relationship to the means of production.
What are the means of production?
Well Marx defines means of production explicitly in his master work,"Das Kaptial"
"If we look at the whole process from the point of view of its result, the product, it is plain that both the instruments and object of labor are means of production"(2)
So in other words, the means of production are what work is done on, and the things you use to do work with. Which means class is defined by your relationship to said items.
Marx defined three classes of people
1 ) The Bourgeois.
The Bourgeois(Also known as Capitalists) have a few distinct characteristics.
A) They own means of production
B) They do not work the means of production
C) Their income is generated through their ownership of capital(I'll go into what capital is later)
2 ) The Proletariat
The Proletariat(also know as the working class) also has a few defining characteristics
A) They do not own means of production
B) They sell their labor power on the market(as a commodity)
C) They do not generate any income from capital
3 ) The Petite Bourgeois
The defining features of a Petite Bourgeois(also know as the middle class) are
A) They own some means of production
B) They still work
C) They own some capital
So why would the proletariat need liberation?
The answer lies in the nature of capital.
Capital is a form of money(marx calls money the universal commodity) that is in motion. In other words capital is money, that has the capacity to increase itself.
What does this mean?
I'll explain.
To first consider capital, we must first understand basic commodity exchange, which is fairly straight forward
So basic commodity exchange looks like this
Commodity->money->Commodity
The example Marx gives in,"Das Kapital"
A person has some linen, they then take it to the market where he gets some money, they then use the money to purchase a bible.
Simple and straight forward.
Capital exchange goes like this
Money->commodity->money
So let us analyze this.
So, a person with money buys, oh i don't know, an iron ingot, for $10. he then waits until the price of iron ingot goes to $12(I guess he could just sell it when the price is lower, but then I guess that would defeat the purpose) he then sells it at the market for $12 generating what is know as,"Surplus value". To calculate the surplus value of simple capital exchange is straight forward. you subtract initial investment from the final exchange-value.
So in this example it would be $12-$10=$2
The two dollars would then be surplus value.
Now capital can get extremely complex, this is just capital in its simplest form. I'll examine capital as a mode of production in a moment.
So this raises some question about the nature of the capitalist system, since, by definition, capitalism is a system based upon captial. Capital, in its simplest form, is a method of increasing wealth without the generation of a use-value(by the person whose wealth capital exchange has increased).
The issue inherit in this simple form of capital is that the wealth of an individual has increased, without the individual generating anything. Right, so there is the contradiction. You can increase your wealth, WITHOUT, working for it, WITHOUT, doing any labor, or anything useful, AND, our entire economic order is based upon this.
Okay, lets examine a capitalist firm, as it would look in the reality which we live.
This is too complex for a simple flow chart so I'm just going to describe it.
So we begin with capital buying of raw materials. Now what are raw materials?
Marx explains in Das Kapital,"All raw material is an object of labor, but not every object of labor is raw material; the object of labor only counts as raw material only when it has gone through some alterations by the means of labor"(3) So raw materials are anything you are working on, that someone has worked on before, like minerals mined from the earth. They are considered raw material because labor was needed to extract them.
Anyway, once raw materials are bought(the first commodity in capital exchange) Labor is also bought from the labor market(Finally the proletariat enters the equation). Meaning the labor power is indeed a commodity in the capitalist mode of production. With this, workers are estranged from their labor. Meaning, that once you sell you labor to someone else, you are no longer the owner of your own labor. Which goes in start contrast to the classic right winger rhetoric claiming that capitalism is when you keep the products of your labor. We've all heard it, and it's objectively wrong.
Anyway, once your alienated from your work, which is an expression of your essence, you are put to the task of working the means of production that someone else owns. So the worker then has his labor,"Absorbed" by the raw materials that then generates a new use value, which is then sold on the market, as a commodity, for a certain amount of money.
So, it follows, that any capitalist who is worth his salt, has figured out a way to sell the new product, for more money then he put in. Meaning that the capitalist who first invested into production comes out wealthier, without actually producing anything with his own hands.
Why is this even an issue? Well as Marx says(in the first sentance of Das Kaptial),"the wealth of societies in which the capitalist mode of production is dominate, is expressed as a vast collection of commodities". Since the proletariat is the producer of society, why should they give up anything to those who do nothing but shuffle around money? Why are the producers(the sweat shop workers, child slaves, etc) paid pennies or nothing, while a few people become richer then is conceivable?
So what does marx purpose instead of private ownership of the means of production?
Well, he says it better then I can
"Let us finally imagine, for a change, an association of free men, working with the means of production held in common"
So who is a Marxist? A Marxist is someone who will analysis things through the lens of economics classes, and one who seeks the liberation of the proletariat from the exploitation of the parasitic class of capitalists.
We can build a better world, from the ashes of the old.
https://www.marxists.org/archive/marx/works/1847/11/prin-com.htm
Das Kapital volume 1, Karl Marx, The Labour process and the valorization process pg.287
3)Das Kapital volume 1, Karl Marx, The Labour process and the valorization process pg.284