A hierarchy is a power relationship; that is, where someone has more power than another, whether through some notional authority, a forceful personality or through access to information and resources. This imbalance is used to dominate, whether it’s an abusive partner, a boss in a workplace, or imperialism. Society also has many hierarchical power relationships founded on discrimination and prejudice.
Even when hierarchy seems benign, there remains the fact that someone has the power to deny someone else access to self-determination, resources, jobs and so on. This is not freedom, no matter how much western culture dresses it up as such. Where there is a threat of losing your job, where your sexuality or skin colour is used to keep you in a weaker position, where you have less determination over your life or access to resources, then there is a hierarchy.
A starting point to challenge this is to call for equal opportunities, for anti-discrimination policies, for equal pay. But it is just a starting point, as a system or society that permits this sort of inequality in the first place is fundamentally flawed. We need to be careful that we do not replicate structures that will allow this sort of oppression to re-establish itself in a different form (eg. right of women to work, but only within a patriarchal system), or simply create a slightly less discriminatory society.
Anything which denies freedom or equality, economic or otherwise, sets up a hierarchy through the tool of discrimination. Often the hierarchies are insidious, built into the fabric of our culture.
This is why saying that you are not homophobic or racist simply because you subscribe to anarchism is not sufficient. It does not deal with the wider context of the society in which you have grown and developed and whose norms you have been constantly exposed to. It takes great arrogance to believe that one is completely free of society’s influence simply because one has declared it. Self-examination is required, and that includes being continually being open to the demands presented by less powerful groups.
So, anarchy is against hierarchy. From this starting point it is not hard to make the leap to anti-capitalism, anti-government and anti-authoritarianism. Militant anti-fascism is also an expression of this approach by anarchists, who recognise the dangers inherent in allowing extreme right wing politics to get deeper footholds in society.
Organised religion is included in this critique, as it tends to be authoritarian and proscriptive, especially in regard to religious leaders.
There is a similar rejection of the traditional left for its centralizing and authoritarian approach. Indeed, it is common in green anarchist thought to emphasis the decentralised approach as the most non-hierarchical.