Anarchism emerged as a modern political movement as a tendency in the socialist movement in the 19th century. Anarchism advocates a particular form of socialism, based on the concept of positive liberty or self-management, where people control decisions to the extent they are affected by them. This means anarchism is opposed to institutions based on top-down power hierarchies such as the governmental apparatus or the corporations. Thus anarchism is anti-capitalist as well as being anti-state. Like socialism anarchism has its origin in working class radicalism.
Anarchism proposes to replace capitalism with an economy based around workers self-management of industry, common social ownership of the means of production, and replacing the state with direct democracy of assemblies, extended to governance over larger areas through delegates who are closely controlled by the base assemblies. The calls for "limited government" by Republicans and socalled "libertarians" are often deceitful, since in reality they favor a powerful state to protect corporate investments throughout the world, and thus a huge U.S. military, and police forces to protect capitalist property against workers or tenants or masses of poorer people.
The forms of government the right-wing want limited are those programs that benefit working class people, such as unemployment benefits, food stamps, minimum wage laws, public transit subsidies, etc. However, anarchists do not favor attacks on the social wage, as these are simply a part of the value created by workers that we have forced the state to give back, through mass struggles in the past. And the same is true for things like free or public health care. Anarchists also propose that people's needs be provided for through systems of social provision such as free health care, free education and so on. Free health care & free education can be argued for on the basis of positive liberty, the roughly equal access of everyone to the means to develop & sustain their abilities.
However, anarchists also propose that public goods be provided not thru the state as at present but through public services managed by workers, and accountable to the direct democracy of popular assemblies. Being against the state as an institution is not the same as being against the services that at present happen to be organized through the state, as Kropotkin pointed out long ago.
There have come to exist since the '60s various small minorities who have tried to appropriate the "anarchist" label despite having right wing politics. On examination, however, it will be found they violate basic ideas of traditional anarchism. For example, "anarcho-capitalists" say they are against the state but they are really against democracy, whereas anarchism historically has advocated direct democrcy.
"Anarcho-capitalists" propose to privatize state functions like police & the judiciary, but the private security forces & arbitration agencies, being run top down & controlled by the wealthy, will in fact be a state. This would be a return to feudalism where the state functions were under private control. The state would still exist because the overall governance functions would be concentrated in certain (private) bureaucracies, separate from popular control. It is this separation which enables the state to serve the interests of dominating, exploiting classes. As to "national anarchists", they are fascists with a racist separatist ideology that violates the traditional anarchist ideas of internationalism & solidarity. Just as the Nazis were not socialists despite the word "socialist" in their name, anarcho-capitalists & national anarchists are not anarchists at all.