What are the rights that people have in anarchy?

I guess this depends on what you mean by "anarchy." In the colloquial sense of chaos and disorder, there are no rights. The "anarchy" to which most anarchists refer, however, is a society in which people have a liberty that is consistent with natural rights, even if they aren't supporters of deontology or natural rights. "Anarchy," in this case, is the absence of rule, be it a semi-permanent institution like the government or individual transgressions upon one's life or liberty. The main difference between an anarchist society and the liberal democracies with which many of us are familiar, aside from the absence of the state, is private property. Historically, anarchists are socialists and therefore don't accept the legitimacy of private property, though a relatively recent group of anarchists support laissez-faire capitalism (anarcho-capitalists). For social anarchists (aka socialist anarchists, left anarchists, libertarian socialists, left libertarians), Natural Resources cannot be owned as private property, but used in common by usufructuaries. This would result in laborers being permitted to use the means of production to create personal property, which they can use or trade as they see fit. Libertarian socialists also want the workplace to be democratic instead of hierarchical, and prefer industries to be managed by worker cooperatives (see syndicalism) instead of bosses whose will must be followed merely because they have the legal right of ownership in private property.