Means of production
The means of production has two broad categories of objects: instruments of labour (tools, factories, infrastructure, etc.) and subjects of labor (natural resources and raw materials). If creating a good, people operate on the subjects of labor, using the instruments of labor, to create a product; or, stated another way, labour acting on the means of production creates a good.
In an agrarian society the means of production is the soil and the shovel. In an industrial society it is the mines and the factories, and in a knowledge economy the offices and computers. In the broad sense, the "means of production" includes the "means of distribution" such as stores, the internet and railroads.
Ownership of the means of production and control over the surplus product generated by their operation is a key factor in categorizing different economic systems. In classical economics the means of production is the "factors of production" minus financial capital and minus human capital.
Factors of production are defined by German economist Karl Marx in his book Das Kapital as labour, subjects of labor, and instruments of labor; i.e., the term is equivalent to means of production plus labor. The factors of production are often listed in economic writings derived from the classical school as "land, labour and capital". Marx sometimes used the term "productive forces" equivalently with "factors of production;" in Capital, he uses "factors of production," in his famous Preface to the Critique of Political Economy, he uses "productive forces" (note that this is in the English versions and may depend on the translation.)
Production relations (Marx: Produktionsverhältnis) are the relations humans enter into with each other in using the means of production to produce. Examples of such relations are employer/employee, buyer/seller, the technical division of labour in a factory, and property relations.
Mode of production (Marx: Produktionsweise)
(In political context) - the facilities and resources for producing goods. Study of how people take over other governments using force and violence, not talks and agreements.
Marxian economic analysis
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The analysis of the ownership and control of the means of production, as well as the level of their technological sophistication, is a central component of Marxist theory.
In Marxist theory, the process of socioeconomic change and evolution is based on the premise of technological improvements in the means of production. As the level of technology improves, existing forms of social relations become inconsistent and unnecessary, creating contradictions between the level of technology in the means of production and how society is organized socially. These contradictions manifest themselves in the form of class conflicts, which develop to a point where the existing mode of production is no longer sustainable and collapses, leading to the emergence of a new mode of production based on a different set of social relations - most notable based on a different form of ownership over the means of production.
Ownership of the means of production and control over the surplus product generated by their operation is the fundamental factor in delineating different economic systems (or modes of production). Capitalism is defined as private ownership and control over the means of production, where the surplus product becomes a source of unearned income for its owners. By contrast, socialism is defined as public ownership of the means of production so that the surplus product accrues to society at large.
To the question of why classes exist in human societies in the first place, Karl Marx offered an historical explanation that it was the cultural practice of Ownership of the Means of Production that gives rise to them. This explanation differs dramatically from other explanations based on "differences in ability" between individuals or on religious or political affiliations giving rise to castes. This explanation is consistent with the bulk of Marxist theory in which Politics and Religion are seen as mere outgrowths (superstructures) of the basic underlying economic reality of a people 
- James M. Henslin (2002). Essentials of Sociology. Taylor & Francis US. p. 159.
- Michael Evans, Karl Marx, London, England, 1975. Part II, Chap. 2, sect. a; page 63.
- Flower, B.O. The Arena, Volume 37. The Arena Pub. Co, originally from Princeton University. p. 9
- Marxism.org: Mode of Production.
- Engels Socialism Utopian and Scientific Chapter III Historical Materialism p.74
- Institute of Economics of the Academy of Sciences of the U.S.S.R. (1957). Political Economy: A Textbook. London: Lawrence and Wishart.