Capitalism, Socialism and Communism are economic models. They affect who owns what property.
Capitalist — I own the cow. You buy the milk. The govt gets its taxes. Or more generally, everyone owns private property, but the govt can tax us.
Socialist — The govt owns the cow. You buy the milk. You pay the govt. More generally, the govt owns resource production (farms, factories), but people can own private property.
Communist — The govt owns the cow. The govt owns the milk. You’re allowed to drink the milk, but only as much as you really need. Govt owns everything, but everyone shares resources.
Democracy, Anarchy, Democratic Republic, etc. are political models. They affect how decisions are made.
Pure Democracy — three of us vote. Two of us decide. Everyone lives with the decision.
Democratic republic — three of us vote. Two of us pick someone to make decisions for everyone, but the person theoretically acts in our best interests.
Delegatory republic — we don’t vote. Someone else picks who makes decisions for all of us, but the person theoretically acts in our best interests.
Monarchy — no voting. One person decides. People may or may not get to do what they want.
Anarchy — no voting. Everyone does what they want.
Fascist — no voting. One person makes the decision. Nobody does what they want.
Economic models (capitalism, socialism, communism) are entirely independent of the political model. You can have a purely democratic communist state just as easily as you could have a fascist capitalist country.