May 27, 2018:
"Mode of Production" is the mature Marx's scientific object. Informally you can think of it as the set of relationships within which stuff gets made. A blacksmith who manufactures iron nails in medieval France produces within a different mode of production from a proletarian who manufactures iron nails in a factory in contemporary France. Same product, different relationship to the means of production, different labor processes, different relationship to the exchange of the product, different relationship to the surplus value realized when the product is exchanged. Formally, Mode of Production is the unity between productive forces and relations of production, where this unity forms the "economic base" of Marx's famous base/superstructure metaphor. The unity is complex, binding together multiple elements such as labor processes, objects of labor, instruments of labor, agents of labor processes, means of production, and so on in a hierarchical ensemble determined by the means of production. Mode of Production is an abstract concept; modes of production aren't found in reality in pure states, but always in complexly articulated combinations with other modes of production.
Mode of Production makes it possible to think history scientifically. Its secondary concepts — social formation, exploitation, social surplus, class, class struggle and so on — enable scientific analysis of historical change. If history interests you, check out Perry Anderson's Passages from Antiquity to Feudalism, which uses these concepts brilliantly to show how the destruction of the Roman Empire and the slave mode of production happened in different ways in different parts of Europe and the Mediterranean, resulting in regional divisions which persist to this day.