Anarchism: Etymologically, from the Greek: Without government; without ruler. In political philosophy those positions that reject the notion that a society should have a - powerful - government.
The term 'anarchism' is also used for social or political chaos, but that is not meant here. There are several kinds and flavors of political anarchism (see Woodcock) and the first clear statements of political anarchism seem to have been formulated in the late 18th Century by William Godwin, and in the 19th Century by Michael Bakunin and, somewhat more rationally, completely and with less glorification of violence, by Peter Kropotkin. Even so, the rejection of authorities, leaders and governments, and the hope that mankind would be happy without these, probably existed since there were authorities, leaders and governments. For indeed, as Lord Acton observed in 1895: 'All power corrupts, and absolute power corrupts absolutely', and many have seen and felt this before it was summed up so well.
The three basic theoretical objections to anarchism are
(1) that it tends to be based on an irrealistic optimistic estimate of human beings on average;
(2) that complex society needs some form of government and some publicly maintained and enforced system of law and order to work at all; and
(3) that the anarchist analyses of society and human beings are too simple-minded, apart from being too optimistic.
The main practical objection is that wherever anarchism has been seriously tried, it soon ran into difficulties and collapsed. Or alternatively expressed: "If men were capable of having a perfect society, it would have existed a long time". (Hazlitt - I believe. Put differently: Men are not perfect enough to built or maintain anything like a perfect society.)
Even so, what is sound in anarchism is what is sound in classical liberalism:
A fundamental distrust of the power of states and governments and a disbelief in the usefulness and goodness of powerful leaders and authorities, and a concern that human beings should be able to govern themselves, instead of being governed in mass by a few, who indeed, as history shows, and as Acton implied, generally abuse their power.