Y'all might know much of these already, but for posterity ....
Socialism is an economic model, in which industry is owned by every member of the public equally and therefore administered by the government. That government can be democratic, autocratic, or anything else; socialism is just a system they use for their money things. It is built around the promise of
equality.
Capitalism is a very different economic model, in which industry is privately owned and goods and services are traded on the free market. It both requires and promotes a class structure of the rich and poor, management and workers and so on, so it is reliant on the premise of
hierarchy.
Communism is an autocratic system of government that envisioned itself as the natural extreme of socialism; a worldwide commune in which everything and everyone is perfectly equal. Unfortunately for the Communists, it also required an authoritarian dictator to get there, so it ended up being, well, Communism. It is an
autocracy that promises
equality.
Fascism is a form of extreme nationalism developed by Mussolini in the 20s. It says we are the best people, we will regain our former glory, all the bad stuff is someone else's fault, come join my army. It is inherent hierarchical, so the last thing it ever wanted was equality for all. A lot of people listened to Mussolini, and the idea spread. It is an
autocracy that is based on
hierarchy.
Robert Paxton pointed out that one of the characteristics of fascism was that it adapted itself to each society. This is what happened when Hitler adapted it for Germany; he called his party National Socialists, not because they adhered to socialist ideas (the whole 'equality' thing would have turned his stomach) but because it marketed well. It occupies the same place as fascism.
Because fascism/Nazism and capitalism are both hierarchical, they both support social classes and private ownership of industry, but the last thing fascists ever want is a free market. They want to be able to control the industries themselves, so it will benefit their own people, not all of those unclean, unworthy others.
Because both Communism and fascism/Nazism are both autocracies, many people lump them together because they are equally non-democratic and
completely non-libertarian, but they are very different in their views and methods. Communist ideology is built around tearing down social classes, and fascists are all about putting themselves alone at the top of the social classes. The last thing Mussolini, Hitler, etc. ever want is to be on the same level as the "others."
Probably the easiest, pop-culture-friendly way to think of the differences between Communists and fascists is to think of them as Klingons and Romulans. They both hate the Federation, but they also can't stand each other.