From a Marxist point of view, an economic system is defined by its mode of production, which is in turn the combination of productive forces and relations of production. Productive forces are, in the words of a Smithian economist, just the labor, land and capital used for production.
What is more important here are the relations of production. The relations of production are the sum of the social relations that define ownership and distribution, often codified in law: for example, the slave owner's proprietary ownership of slaves, the feudal lord's relationship with his fiefs and serfs, and the capitalist's private ownership of the capital used for production. Marx proposed that, over human history, the development of productive forces creates a strain on the existing relations of production, and the conflict between them causes the relations of production and thus the mode of production to evolve.
Marx described a few modes of production, i.e. economic systems, that existed in human history in his Critique of Political Economy and Das Kapital:
- Primitive communism. In a primitive communist economic system, production is so low that no permanent surplus product, i.e. production beyond what is required to sustain everyone, is produced. Consequentially, there is no concept of ownership in this a primitive communist economy, because there is nothing over which it is possible to contend ownership. Because of the lack of any private property, there are no classes in a primitive communist society.
- Ancient mode of production, aka slave society. When forces of production develop such that some permanent surplus is produced, people will naturally begin to contest for this surplus. People then realize that directly controlling the laborers who are in turn able to produce surplus is the most effective way, because at this level of development human labor is the determining force of production. Thus, a class of slave owners begin to directly control a class of slaves, who in turn produce surplus for the slave owners.
- Feudalism. When forces of production further develop, arable land will become more significant. Feudalism, therefore, is such a system that allows effective control of arable land. In a feudal economy, feudal lords control serfs contingent upon land. The serfs work on the land and give all the surplus to their lord, in return for what is necessary for a living, protection, etc.
- Capitalism. With the rise of skilled & waged labor, development of division of labor and manufactories, capital will become the foremost important resource in production. Thus, in a capitalist economy, private property, i.e. private ownership and control of capital essential for production. A class of capital owners, better known as the bourgeoisie, employ workers, or proletarians. The bourgeoisie owns the capital and thus controls all the surplus, and the proletarians are given a monetary wage for their work. This is the economic system that best characterizes most of the world at the current stage.
The rest of the modes of production are, by far, just theory:
Socialism. Marx predicted that capitalism will have its problems. The mass accumulation of capital will become less profitable due to the law of falling returns. Moreover, the development of capitalism will create an ever-growing class of proletarians, who find their life to be more difficult than ever. Eventually, the class of bourgeoisie is overthrown, and all producers direct their production according to need and distribute their production by contribution. There haven't yet been a fully socialist economy in human history yet, but we are seeing modes of production that approach socialism, often with the intervention of a government. Examples include:
- central planning, which is the Soviet Union's failed attempt at socialism. The government plans productions according to their perception of society's needs, and controls distribution, in principle according to contribution. We all know why this failed.
- social democracy, which attempt to incentivize production according to need and make distribution more close to distribution by contribution by utilizing governmental intervention in the economy, such as a graduated income tax, public ownership of corporations, and governmental subsidies, etc.
- cooperatives, in which a group of people democratically discuss their production and the method to distribute their production, often by contribution.
Communism, the ultimate mode of production according to Marx. When production is so efficient and material wealth is so abundant, everyone's needs could be eventually fulfilled. Thus, distribution by need is possible, and everyone could get what they need.
The modes of production aren't discrete. There is always stages "in the middle". Social democracy and cooperative economy, as I mentioned above, are examples of transitional modes of production between capitalism and socialism, and are widespread in the world today.
If you are interested in Marxist theories of modes of production, you could read more about Marx. Diving right into Das Kapital will be difficult, but beginning with shorter works such as The Communist Manifesto and Critique of the Gotha Program, etc.