Simple Explanations of Political Systems
In common usage, the word capitalism means an economic system in which all or most of the means of production are privately owned and operated, and the investment of capital and the production, distribution and prices of commodities (goods and services) are determined mainly in a free market, rather than by the state. In capitalism, the means of production are generally operated for profit. In a purely capitalist economy, there would be no public schools, no state owned or maintained roads and highways, public works, welfare, unemployment insurance, workers compensation, Social Security benefits etc.
Most generally, socialism refers to state ownership of common property, or state ownership of the means of production. A purely socialist state would be one in which the state owns and operates the means of production. However, nearly all modern capitalist countries combine socialism and capitalism. The University of Idaho, and any other public school or university, is a “socialist” institutions, and those who attend it or work for it are partaking in socialism, because it is owned and operated by the state of Idaho. The same is true of federal and state highways, federal and state parks, harbours etc.
Most generally, communism refers to community ownership of property, with the end goal being complete social equality via economic equality. Communism is generally seen by communist countries as an idealized utopian economic and social state that the country as a whole is working toward; that is to say that pure communism is the ideal that the People’s Republic of China is (was?) working toward. Such an ideal often justifies means (such as authoritarianism or totalitarianism) that are not themselves communist ideals. Fundamentally, communism argues that all labour belongs to the individual labourer; no man can own another man's body, and therefore each man owns his own labour. In this model all "profit" actually belongs in part to the labourer, not, or not just, those who control the means of production, such as the business or factory owner. Profit that is not shared with the labourer, therefore, is considered inherently exploitive.
The word descends from the Latin ‘fasces’, the bundle of sticks used by the Romans to symbolize their empire. This should clue you in that Fascism attempts to recapture both the glory and social organization of Rome. Most generally, “a governmental system led by a dictator having complete power, forcibly suppressing opposition and criticism, regimenting all industry, commerce, etc., and emphasizing an aggressive nationalism and often racism.” Unlike communism, fascism is opposed to state ownership of capital and economic equality is not a principle or goal. During the 1930s and WWII, communism and fascism represented the extreme left and right, respectively, in European politics. Hitler justified both Nazi anti-Semitism and dictatorship largely on the basis of his working to fight-off communism.