Socialism proper refers to the advocacy of some sort of collective, social, or public ownership of industry/enterprise and, usually, of land as well. We find many different varieties of socialism. Ricardian socialists advocate replacing private ownership of the means of production with worker-owned cooperatives. Marxists advocate public ownership by the State, which in turn would be run by the proletariat. Lange-Lerner socialism advocates public ownership of enterprise within a market system with central planning. Then there are the libertarian socialists, like Proudhon and Bakunin, who advocated worker-owned cooperatives alongside public ownership of land at the municipal level. Kropotkin’s variation of libertarian socialism had communal ownership, like decentralized Marxism minus the State, rather than independent worker-owned cooperatives as with Proudhon's mutualism and Bakunin’s collectivism. But all varieties of socialism have one common characteristic: they oppose private ownership of the means of production in favor of some sort of social ownership.
Capitalism, unlike socialism, refers more to the condition of things than the precise form of ownership. Capitalism is not private ownership and markets. There are countless examples of market economies with private enterprise that we would not regard as capitalism. Feudalism is a key example, as was medieval distributism. An essential characteristic of capitalism is the division of society into two classes, haves and have-nots, where the have-nots must work for the haves as wage-laborers in order to survive. Under capitalism, most people do not have any share of private ownership over the means of production, meaning that a small group of capitalists and landlords essentially rule over the masses. The vast majority of people under capitalism are not capitalists, which means that they must sell their labor to the capitalist class in order to survive. Wage-slavery is an essential characteristic of capitalism. For a system to be capitalism, it must have money, markets, and private ownership. It must be industrialized. Those are necessary but not sufficient conditions for classifying a system as capitalism. To be capitalism, wage labor must also be the normal mode of survival. Markets, private ownership of land and enterprise, and wage-slavery are the three necessary conditions that must be met in order for a system to constitute capitalism. There are other systems that also have division of people into haves and have-nots but do not constitute capitalism. Feudalism may have markets, private property, and division into haves and have-nots, but it doesn't have wage-labor as the norm and does not have an industrialized economy, so it is not capitalism.