Left anarchism, which was the common usage of the term up until recently, is a political philosophy very closely related to communism, with the central socialist principle that workers should own the means of production. Having bosses run businesses or capitalists own the means of production is considered to be exploitative and hierarchical, whereas right anarchists see these relationships as acceptable so long as they are voluntary: if employees can agree to contractual terms, quit any time they like, unionise except where prohibited by voluntary agreements, etc. it is considered that the bosses are providing value by organising the work force, and capitalists are providing value by anticipating the market, organising production in terms of demand, and risking their investment (and also their property where uninsured, as the state would not exist to give them limited liability in the form of a Ltd. corporation or Plc.) Right anarchists are not opposed to workers owning their own workplaces as long as this is organised by mutual agreements and voluntary interactions rather than imposed by violations of property rights (eg. workers getting a loan from the bank to buy out bosses and owners.)
Right anarchism is a more recent development but it can also be traced back to Baukunin who is seen as influential to both schools as an anarchist. It's more of an extension of American libertarianism, with the central principle being the NAP - ie. Thou shalt not initiate the use of force - no violence or theft, including taxation, as a rule, although levies for public goods could be extracted through a system of social pressure, ostracism, and refusal to engage in collective buying for people who did not pay their share (eg. If you don't pay your share for street lights you can expect to have to make your own arrangements for garbage collection as well.) Government regulation would similarly be be replaced by a system of insurance companies or cooperatives (often referred to as DROs - dispute resolution organisations) who were financially incentivized to solve problems before they occurred instead of being called upon to respond after the fact (like police, government fines, universal sick care, fire fighters, climate taxes etc. and all government services which are called upon only when a problem has all ready arisen.) These companies would lose money through payouts when they failed to protect their clients from harm or loss of property, which would encourage them to develop preventative measures and disincentives to criminals which would be constantly optimised through competition on the free market - whichever organisation was most effective as preventing harm or loss would gain the greatest market share, if someone advanced on their developments they would become lucrative, and if any such company got "too big for its boots," or abused its authority, its clients would have the option to pick another service provider, rather than remain at the behest of a state monopoly for provision of this service.
Left anarchism tends to focus more on problems of capital and capitalism, much like state socialists do. Right anarchists tend to talk primarily about the problems of statism, those which are created or exacerbated by governemnt, and they therefor tend to have the most convincing arguments for the abolition of states.
Despite coming from different angles, both agree that the state is based on force or the threat of force for its existence, that the state is a tool wielded by the ruling class for unfair advantage over everyone else, and that corporations and corporatism are products of statism which allow the rich to privilege from privatizing gains and socialising losses, which is immoral.
All forms of anarchists acknowledge that where there are rulers there can be no rules, as rulers by their very nature, make themselves the exception to the rule. Anarchists are egalitarians, no special privileges for law-makers or corporations. If you kill, harm, steal, or damage someone else's property, you are to be held personally liable for it - not the state (tax payer) or corporation (consumer/share-holder) - you personally. There is to be no hiding behind institutions which are mere abstractions of the mind in anarchy. This implementation of this moral hazard for the privileged is meant to do away with much of the corruption, cronyism, and war crimes which are part and parcel of statist societies.
So some common ground between the two despite philosophical difference on very key points.
For example, left wing ideals such as workers running their workplaces are thought by some to be more likely under right anarchism that statism, since the public education system trains children for individualism and competition, but the evidence on how individuals learn best is in favour of a cooperative learning environment. Without the kind of schooling which is prevalent and imposed by the state (which right anarchists are strongly opposed to) children would be raised with lots of experience of cooperation and mutualism, and so would more likely to create workplaces that capitalised on those skills than top-down hierarchies which follow the prevalent pattern of schooling and parenting.
