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When studying the etymology of the word anarchism we find that the root is Greek - αναρχία - and means "without rules/ruler".
Anarchism is a political ideology devided in numerous branches with various protocols and desires. What unites anarchists from different branches is a vision of a stateless, non-heirarchal society where people are free to choose if, and to what extent, they wish to contribute for the sake of community. The main directive is freedom for the individual. As mentioned, there are many branches within anarchism and some of them are more focused on individuality while other are more into collectivism. It is fair to say anarchism is a far-left ideology, often associated with communism, collectivism and syndicalism. There is no fixed body of doctrine and the overall philosophy is fluctuant.
Even though anarchist themes can be found as early as in 600 BC, in the works of Taoist philosopher Laozi, anarchy was officially named and formed as a political and philosophical ideology back in the mid 19th century. The father of anarchy is considered to be Pierre-Jospeh Proudhon. Another famous anarhist was Jesus - one of many anarchist branches is the Christian Anarchism.
Libertarianism has often been used as a synonym for anarchism, and has possibly had a better global spread. The two words are not synonyms in all contexts though, because sometimes libertarianism strictly refers to a individualistic free market philosophy.