Compare capitalism, socialism, and communism as to (a) how each allocated scarce resources among alternative wants and (b) the existence of private property.

In a purely capitalistic economy, resources are allocated based on the interests of both suppliers and consumers. The allocation of resources should be more efficient since the ultimate goal of businesses is to make a profit by satisfying consumers' preferences. Private property is one of the fundamental pillars of capitalism.  

In a socialist economy, an elected government is responsible for allocating resources that should best satisfy the needs of the people that they rule over. Socialistic theories are supposed to be a superior state of capitalism, where everybody should share and enjoy resources but based on their individual contributions. I.e. if you work harder, you should be able to enjoy more resources. Socialist theories recognize individual private property but productive capacity is a common good. E.g. the world in Start Trek is a socialist world.

Communism shares several similarities with socialism, but it differs in some of its basic principles. Communism does not recognize elected governments, e.g. China, North Korea, etc., or have pseudo democracies like the former Soviet Union (currently Russia) and Cuba. Resources are shared by everyone but based on individual needs, not individual contributions. E.g. a person that doesn't work at all could need more things than others that work, and therefore, should receive more resources. Everything is owned and distributed by the government, so theoretically private property doesn't exist. Communism abolishes religion and social classes, while socialism allows religion and still recognizes different social classes.

In real life, there is no purely capitalistic or purely socialist country in the world. Some developed economies lean towards socialism (most European nations and Canada) while others lean more towards capitalism (the US). Communism used to exist in the world, e.g. former Soviet Union, China, Cuba, etc., but even those countries currently recognize private property, although the governments still play a significant role in the economy and resource allocation. But they cannot be considered democracies either.