Marxism can roughly be defined as the application of the materialist method to the study of social phenomena. It regards the material conditions and class conflict as driving forces behind historical social development -- Marxism stresses the causal effects of material conditions and class conflict. Marxism primarily focuses on explaining society and history.
It is difficult to establish a precise definition and framing of Marxism can depend on the emphasis preferred by a theoretician on one aspect of the method over another. The complexity of Marx's writings allows for misreadings, misinterpretations, and misplacing emphasis, and enables abuse. Arguably, Marxism as comprehensive world view originated after Marx's death from the efforts of Karl Kautsky, Friedrich Engels, and Georgi Plekhanov. Classical Marxism is based on the writings of Karl Marx and Friedrich Engels.
Marxism is not an ideology. It is strictly a method of social analysis. Scientific socialism is a socialist political ideology based around Marxist theory.
Placing different emphasis on different aspects of Marxist analysis produces different branches of Marxism. Different schools place a greater emphasis on certain aspects of classical Marxism while de-emphasizing or rejecting other aspects of Marxism, sometimes combining Marxist analysis with non-Marxian concepts. Some variants of Marxism primarily focus on one aspect of Marxism as the determining force in social development – such as the mode of production, class, power-relationships or property ownership – while arguing other aspects are less important or current research makes them irrelevant. Despite sharing similar premises, different schools of Marxism might reach contradictory conclusions from each other. For instance, different Marxian economists have contradictory explanations of economic crisis and different predictions for the outcome of such crises. Furthermore, different variants of Marxism apply Marxist analysis to study different aspects of society (e.g. mass culture, economic crises, or sexism).
Marxist understandings of history and of society have been adopted by academics in the disciplines of archaeology and anthropology, media studies, political science, theatre, history, sociology, art history and art theory, cultural studies, education, economics, geography, literary criticism, aesthetics, critical psychology, and philosophy.
Marxism is derived from the name of German philosopher Karl Marx. The term was popularized by Karl Kautsky who considered himself an "orthodox" Marxist during the dispute between the orthodox and revisionist followers of Marx. Kautsky's revisionist rival Eduard Bernstein also later adopted use of the term. Engels did not support the use of the term "Marxism" to describe either Marx's or his views. Engels claimed that the term was being abusively used as a rhetorical qualifier by those attempting to cast themselves as "real" followers of Marx while casting others in different terms, such as "Lassallians". In 1882, Engels claimed that Marx had criticized self-proclaimed "Marxist" Paul Lafargue, by saying that if Lafargue's views were considered "Marxist", then "One thing is certain and that is that I am not a Marxist".
Marxist analysis begins with an analysis of material conditions, taking at its starting point the necessary economic activities required by human society to provide for its material needs. The form of economic organization, or mode of production, is understood to be the basis from which the majority of other social phenomena – including social relations, political and legal systems, morality and ideology – arise (or at the least by which they are directly influenced). These social relations base the economic system and the economic system forms the superstructure. As the forces of production, most notably technology, improve, existing forms of social organization become inefficient and stifle further progress. As Karl Marx observed: "At a certain stage of development, the material productive forces of society come into conflict with the existing relations of production or_ this merely expresses the same thing in legal terms with the property relations within the framework of which they have operated hitherto. From forms of development of the productive forces these relations turn into their fetters. Then begins an era of social revolution.
These inefficiencies manifest themselves as social contradictions in society in the form of class struggle. Under the capitalist mode of production, this struggle materializes between the minority (the bourgeoisie) who own the means of production, and the vast majority of the population (the proletariat) who produce goods and services. Taking the idea that social change occurs because of the struggle between different classes within society who are under contradiction against each other, leads the Marxist analysis to the conclusion that capitalism exploits and oppresses the proletariat, which leads to a proletarian revolution.
Capitalism (according to Marxist theory) can no longer sustain the living standards of the population due to its need to compensate for falling rates of profit by driving down wages, cutting social benefits and pursuing military aggression. The socialist system would succeed capitalism as humanity's mode of production through workers' revolution. According to Marxism, especially arising from Crisis theory, Socialism is a historical necessity (but not an inevitability).
In a socialist society private property in the means of production would be superseded by co-operative ownership. A socialist economy would not base production on the creation of private profits, but would instead base production and economic activity on the criteria of satisfying human needs – that is, production would be carried out directly for use. As Engels observed: "Then the capitalist mode of appropriation in which the product enslaves first the producer, and then appropriator, is replaced by the mode of appropriation of the product that is based upon the nature of the modern means of production; upon the one hand, direct social appropriation, as means to the maintenance and extension of production_ on the other, direct individual appropriation, as means of subsistence and of enjoyment."
The Law of Value
Revolution and Dictatorship
Engels' role in the development of Marxism:
I cannot deny that both before and during my forty years' collaboration with Marx I had a certain independent share in laying the foundations of the theory, but the greater part of its leading basic principles belongs to Marx ... Marx was a genius; we others were at best talented. Without him the theory would not be by far what it is today. It therefore rightly bears his name. 
- Letter to Bernstein, 1882.
- A Great Socialist—Frederick Engels