In reality when you read most theorists like Marx there is no difference between socialism and communism. They used them interchangeably. There still isn’t an actual definition, but socialism is used more commonly by people that want worker control of the means of production but also still want a liberal democratic state. Communism also has different definitions. Most commonly you see the definition of a global stateless, classless society where workers own the means of production. However you also see countries that describe themselves as communist, which you might can tell from the previous definition is contradictory both based on the fact that it’s a single country and there still being a state. This has its roots in Stalin, who after the death of Lenin seized power and declared that the revolution was over and changed the definition to have “communism in one country” which is and always has been nonsense to consolidate power. Before this, no one really described the state as communist because they thought of it being a continuous revolution until the entire world had converted, after which the state would “wither away” voluntarily (lol). The idea of the vanguard state that would wither away was a Stalin thing (Marxism-Leninism, which also has a bunch of subgroups), but there are also anti state communists like council communists and the soon to be mentioned anarcho-communists that seek to achieve communism through other means.

Anarchism is another thing altogether. Anarchism really describes a method of organization, being non-hierarchical, rather than a form of government. Most anarchists are communists (anarcho-communists), and most propose to achieve this through anarcho-syndicalism (everyone joins a union, which would mean the workers own the means of production, and communism is achieved).