I've written a simplified explanation of the difference between communism, oligarchy and socialism, I'll print it here, do feel free to share it if you find it useful.

Explaining the difference between oligarchy, socialism and communism, the short version:
Oligarchy assumes that most, if not all the profits should go to the owners, investors and managers of a business. Such a society will tend to cut taxes on the rich, who control government, leaving lower wage earners to pick up the slack.
Communism assumes that most, if not all of the profits should go to the workers who create those profits through their labors. However it does this by having the State taking ownership of the businesses, which then pays the workforce wages, in theory with everyone more or less earning the same wage as everyone else. No need for taxation here as the bulk of the profits go to the government.
Socialism assumes that everyone involved in a business is contributing towards its success, therefore everyone involved from the CEO to the janitor should get their fair share of the pie.
Socialism it should be pointed out still allows for higher pay for CEO's, etc, for example, it just doesn't let them have the whole pie because they didn't do it all by themselves. That's why socialists value and discuss concepts such as social justice, things like allowing a full time worker to earn a living wage because no business has the right to exist if it cannot, or will not pay an honest days wage for an honest days work. In other words, they are trying to create a just society, in which higher wage earners pay higher taxes and lower wage earners pay lower taxes, which taxes are then used to fund such aspects of national infrastructure as education, health care, emergency services, roads, national defense and so on. Quite a reasonable goal and objective if you ask me.