The questioning of the status quo of the economic system was not a new idea; Jean-Jacques Rousseau , with his famous "Man is born free but everywhere is in chains",  sowed its first seeds and was followed by a plethora of thinkers whose ideas are better described in the article Economic Heterodox Tradition. Each one of those philosopher-economists, during the XVIII-th and early XIX-th centuries, pointed out some of the flaws they identified in the prevailing economic system. Their rethoric was strong - Proudhon would proclaim: "property is theft"  - and such arguments gathered some large audiences; from time to time they would succeed in promoting big riots and even some short lived revolutions . But none of those thinkers developed a systematic and encompassing political economy theory framework which had the strength to possibly change the world.
Some of them were too inconsistent, or too utopic; some were too romantic, or too idealistic, or even "half-mad". Most of the experiments which tried to implement their theories in the real world did not last long. Even those which partially succeeded, like Robert Owen's (1771-1858)  example of the viability of co-operative factory communities, the "New Lanark Mills"  in Scotland, represented only punctual local experiments, unable to influence the economic activities beyond the limits of their own nearby village.
On October 1st, 1867 the publication of the first volume of a single book would change that situation. The book was Karl Marx's Das Kapital - Kritik der Politschen Ökonomie . A very long, dense, complex, logical, sometimes tedious, but a very systematic work Das Kapital, since its publication, has been read by a very few, but has been discussed by all. Marx died before he could complete the work; his friend and sometimes co-writer Friedrich Engels  edited Das Kapital 's second and third volumes, based on Marx's notes.
The marxian economists have been able to inovate and modify most of Economics but in one aspect: the "natural propensity" of economists to disagree among themselves. Immediately after its birth, marxian economics splat in two main currents: the orthodox, led by Friedrich Engels  (1820-95), Karl Kautsky , Rosa Luxemburg, Georgy Plekhanov and others and the revisionists, led by Bernstein (1850-1932), Jean Jaurès (1859-1914), G.D.H. Cole (1890-1959), Sidney Webb and the Fabian Socialists, Mikhail Ivanovich Tugan-Baranovsky (1865-1919), Werner Sombart and the (Youngest) German Historical School, among others.
Karl Marx  work suggested that the number of conditions required for steady-state growth were too numerous for capitalism to avoid its own breakdown. This "inevitablibilty" of the capitalism failure was challenged by Bernstein (1899)  who thought that if socialism is to exist, it must be a conscious choice.
The orthodox marxian economics
The Marxist School challenged the foundations of Classical theory. Writing during the mid-19th century, Karl Marx saw capitalism as an evolutionary phase in economic development. He believed that capitalism would ultimately destroy itself and be succeeded by a world without private property. Marxist, neoclassical, and Keynesian economics all have their roots in the "Classical Political Economy" of the early XIXth century. One key point to understand is that Marxist economics is an economic theory of capitalism. Another important point to have always in mind is that the "soviet-type" of marxism, that is to say, the implementaion of "marxism" by the governments of the Soviet Union and other communist contemporary coutries bear little relation to the system as devised by Marx; they only represent the "soviet" interpretation of marxism.
The essence of Karl Marx's work is centered on the labour theory of value. Adam Smith thought the amount of labor required to produce the goods would determine the rate at which they could be exchanged for one another. However Smith's theory of value was sometimes inconsistent. David Ricardo elaborated on Smith's theory. Ricardo's ideas, tightly reasoned and complex, were much more than a reconsideration of the Smith's labor theory. They gave directions to economics that it is still following today. A good grasp of David Ricardo's theory of value will prove to be an essential pre-requisite for those willing to understand the principles of marxian economics. ("From the viewpoint of pure economic theory, Karl Marx can be regarded as a minor post-Ricardian". Paul Samuelson,The American Economic Review, March 1962, pp. 12-15)
- The value of a commodity, or the quantity of any other commodity for which it will exchange, depends on the relative quantity of labour which is necessary for its production, and not as the greater or less compensation which is paid for that labour. 
This has been the starting point of Marx's critique. A second ricardian discovery would also drive Marx's analysis further; this had to do with land rent. Land rent seems to be a cost of production. So shouldn't the "natural price" of an agricultural product (for example, potatoes) depend on the rent of land? But labor will be more productive on land that is more fertile. Crops grown on fertile land will cost less labor. Does that mean those potatoes will have less value?
Ricardo discovered that the rent of land would be just enough to offset the differences in labor cost, so that the value of agricultural products (potatoes) would be the same regardless of the fertility of the land where they were produced. That is because rent is based on differential productivity.
From this Ricardo drew two main conclusions:
- 1. It is the labor required for production on marginal land that determines the normal price or value of agricultural products.
- 2. The surplus of production on more fertile land is absorbed by rent. Landowners don't have to do anything to earn this rent -- they get it automatically as a result of the competition for fertile land.
Marx and The Labor Theory
After Ricardo the Classical Political Economists defined that the relative value of two commodities in exchange would be equal to "the relative quantities of labor-time embodied in the two commodities". Values always corresponded to labor. In other words, "labor produces all value". But if labor produces all value, how can there be profits or interest?
For Ricardo landowners would obtain rent without contributing any effort, just because of the workings of the competitive market system and the labor values of products. The landowners were beneficiaries of a surplus-value because they had title to relatively productive land. Marx' idea was that all market payments other than wages -- all profits, interest, and rent -- could be explained in terms of surplus value.
Marx expressed his labor theory of value more precisely than Ricardo. In Marx's terms, "the value of a commodity is the 'socially necessary' labor time embodied in it". This phrase, "socially necessary," clears some minor confusions in the theory:
Suppose Joe is a carpenter, but he is very old, so it takes him twice as long as other carpenters to build a house. Does that mean the houses John builds are worth twice as much? No, Marx would answer: since there are other carpenters who can build the house in half the time, half the time constitutes the "socially necessary" labor time.
The concept "that the value of a commodity is the 'socially necessary' labor time embodied in it" is central to Marxist thinking.
From this principle Marx drew the following conclusions:
- In a competitive capitalist economy, all commodities are priced at their values.
- In a competitive capitalist economy, labor is a commodity.
- Therefore, in a competitive capitalist economy, labor is priced at its value.
From this Marx concluded:
- In a capitalist economy, the "natural price" or value of any commodity is its cost of production.
- Labor, the common element of all production, is the measure of cost.
- It is the socially necessary labor that determines value. Inefficiency does not increase value.
- Labor embodied in machinery is incorporated in the value of the output proportionately as the machinery is used up.
- The value of educated labor will be greater than that of uneducated labor proportionately as labor is required for the education.
- In a capitalist economy, labor is a commodity, and so the value (wage) of labor is the cost of production of labor, measured in labor hours.
- In a modern economy, thanks to increased productivity, a labor-day can produce more than its cost.
- This difference is "surplus-value" and is the basis of capitalist profits and "exploitation".
Exploitation and History
For Marx "capitalism" is a society divided between two classes: the working class (which produces all value) and the owning and employing class (which, without producing anything, exploited the working class, extracting from them the surplus-value). For him, the most important characteristic of capitalism was the class struggle between the working class and the employing class over the appropriation of surplus-value.
Marx recognized that "capitalism" in its time had been a very dynamic system and that it had increased the productivity of labor. However, his experience - of the middle of the nineteenth century capitalism - suggested that capitalism was getting unstable. The Classical Political Economists believed that the rate of profit would tend to decline toward zero, over time; and Marx accepted this. At the same time, it seemed to Marx that the working class were getting poorer. These three tendencies -- increasing instability, a falling profit margin and the inevitable "immiserization" of the working class -- could not go on forever, reasoned Marx. So he predicted that eventually they would lead to a crisis, causing the collapse of the Capitalist system. The working class would be ready to take over power.
It is next to impossible to understand how profit margins could fall and the working class be "immiserized" simultaneously. Marx tried to explain this point by analyzing what he called the "Organic Composition of Capital," . The Organic Composition of Capital is the ratio of the value of the materials and fixed costs (constant capital) embodied in production of a commodity to the value of the labour-power (variable capital) used in making it) . Using this concept, Marx created a complex, difficult to understand and very long explanation (which can be read on the links) . In this article, for the sake of simplicity, we shall adopt some modern neoclassical economists views that "it is hard to understand because it cannot happen".
Marx also tried to justify his predictions of the capitalism collapse based on the marxian theory of history. For him the theory of surplus-value was a theory of history. So, for Marx, a new society would emerge. Capitalism replaced feudalism through a series of revolutions, in which the class of "capitalists" or "merchants" defeated the class of landlords. For Marx the exploiting class requires the working class, but the working class does not require exploiters, and will dispense with them, organizing production for its own benefit as a class. This new society would be the "socialism", a new, post-historical society without exploitation which would be the "marxist socialism", or "socialism" as conceived by Marx himself, also called "mechanical socialism" . "Marxist socialism" supposes a class war, resting on the clear-cut distinction devised by Marx, and conceives a logically developed system of the control of the economy by government.
For Marx the emergence of the new socialist society would be revolutionary:
- ...the forest of uplifted arms demanding work becomes ever thicker, while the arms themselves become ever thinner 
Marx did not antecipate the changes in evolution of capitalism that would prevent this vision to become reality. Or, on Keyne's words: "Marxian Socialism must always remain a portent to the historians of Opinion - how a doctrine so illogical and so dull can have exercised so powerful and enduring an influence over the minds of men, and through them, the events of history." 
- Marxian School
- Friedrich Engels
- Karl Kautsky
- Jean-Jacques Rousseau.Page at The History of Economic Thought Website
- ROUSSEAU, Jean-Jacques. The European Enlightenment.
- PROUDHON, Pierre-Joseph. What is Property? An Inquiry into the Principle of Right and of Government. New York: Dover, 1970. Translated from the French by Benj. R. Tucker. Originally published by Humboldt Publishing Company, 1890..
- 1848: Europe in Revolt. Internet Modern History Sorcebook
- OWEN, Robert Owen. A New View of Society, Or, Essays on the Principle of the Formation of the Human Character, and the Application of the Principle to Practice (1813-16). The Avalon Project at Yale Law School
- New Lanark Mills
- MARX, Karl. Das Kapital - Kritik der Politschen Ökonomie. 1867. First english edition of 1887 (4th German edition changes included as indicated). Translated by Samuel Moore and Edward Aveling, edited by Frederick Engels
- Karl Marx
- RICARDO, David. The Principles of Political Economy and Taxation. 1815 (third ed. 1821). Hamilton, Ontario: McMaster University.
- MARX, Karl,Organic Composition of Capital, inDas Kapital, vol III, ch. 8
- HOBHOUSE, L. T Liberalism (1911), excerts. Internet Modern History Sorcebook
- MARX, Karl. (1849), "Wage Labor and Capital". Robert C. Tucker, ed. pp. 203-217
- KEYNES, John Maynard. The End of Laizzes-faire, in Essays in Persuasion. W W Norton & Co,1991. ISBN 0393001903