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A big part of the confusion comes from the fact that different people will use these terms differently.

In a capitalist framework, there's private property and public property. Either an individual (or or specific group) own something, anything, or it's owned by the government.

In a socialist framework, private property is distinguished from personal property. Personal property is your stuff that you use for yourself. Your coat, your car, your TV, etc. Private property is the means of production, or capital—things that increase a worker's ability to do useful work. Think factories or companies, where ownership in and of itself, regardless of labor, would make the owner money. Socialists think that kind of private property shouldn't exist, because it means wealthy people can just own stuff for a living, profiting off of the people who do the work.

Housing can go either way. Owning a home for yourself and your family would be far closer to personal property, while owning an apartment building to collect rent would be far closer to private property.

Socialism, for the most part and historically, is an umbrella term describing social rather than private ownership. That would include anarchism, which largely synonymous with "libertarian socialism." Lenin, on the other hand, used it to more specifically refer to an intermediate stage between capitalism in communism, so you might see people using that more narrow definition to exclude anarchists, democratic socialists, etc.

A few things draw significant differences.

Anarchism is fundamentally a firm rejection of unjust hierarchy, including the state, via building up of bottom-up structures using networks of Mutual Aid or other strategies (like Syndicalism).

Communism is fundamentally about advancing beyond Capitalism into Socialism and eventually Communism. It's fundamentally Marxist, unlike most forms of Anarchism (which don't necessarily reject Marx, but also don't accept everything Marx wrote). Communists are generally perfectly fine with using the state in order to eventually achieve a Stateless, Classless, Moneyless society, as each becomes unnecessary and whithers away.

In essence, Anarchism rejects that a state is necessary at all, and seeks to directly replace current systems with the end-goal of an Anarchist structure, whereas Communists tend to agree more with gradual change, rapidly building up the productive forces, and achieving a global, international Communism.

Anarcho-Communism seeks to combine these into directly implementing full Communism without going through Socialism first.

All of this is from a generally Leftist perspective, without leaning into any given tendency, as I believe the most critical battles now are building up a sizable leftist coalition. Everyone should focus on organizing, unionizing, reading, learning, sympathizing, empathizing, and improving themselves and those around them.

I've never heard anyone argue against personal property. Usually the difference is that Anarchists want to skip the workers' state, while other Communists think it's a necessity to achieve Communism.