The teeming millions need some pointers here.
Communism is as communism does. It’s details are set forth in the Communist Manifesto by Karl Marx. Also called Marxism. There are suggestions that Marx disavowed this when he said “I am not a Marxist.” For Republicans and Democrats alike, think of Communism as a utopian idea (like Plato’s Republic and others) that just didn’t work out and probably was never intended to. The short version of what Communism is/was: collectivising so that each may contribute as they are best able and have their basic needs met. In short, a publicly controlled economy to benefit each person in basic needs. Needless to say, it ignores human nature and has never worked. It is considered left wing. (Wings, right or left are based on which side of the parliment each faction sits on. “People’s” factions on the left and “aristocracy” on the right, and no, I do not remember which Parliment this was.)
Fascism (the name) comes from an ancient Italian tribe that put the needs of the state (city state at that time, pre-Roman) ahead of any individual interest. The principle is symbolized by a bunch of sticks bound together in something called fasces, which can be seen on the back of the dime (an insult to occupant of the obverse FDR, who was the greatest anti-fascist who ever lived). It is also the symbol of the United States Senate and may be observed behind the President of the Senate’s chair. The short version is that the needs of the state to survive and become great and to which each and every individual is to submit completely and sacrifice themselves. In our ancient Italian tribe, the tribal leader put to death his beloved son for some minor sin against the state.
Fascism in practice is an ultra-nationalistic capitalocracy, in which corporate interests are primarily pursued. It doesn’t tend to work very well. Military dictatorships usually fall into this category.
Nazism is something different entirely. The short version is that Hitler said different things at different times in order to appeal to different constituencies, but with the purpose of achieving absolute power in the state. The name of Hitler’s party was the National Socialist German Worker’s Party (or something like that). It at first appealed to extreme patriotism in the humiliating aftermath of WWI and was originally targeted to the millions of disaffected veterans (of which Hitler was one) and blaming their defeat, humiliation, unemployment and impoverishment on highly unpopular, voiceless and powerless social groups (who weren’t in a position to fight back), namely the Jews and others. Once Hitler’s movement gathered steam, me met privately with industrial leaders and assured them that he would not “nationalize” their businesses, but rather re-arm (incidentally in violation of the Treaty of Versailles) and make them incredibly wealthy, and break all unions. In short, secretly selling out the socialist part of the movement. In quick order, Nazism became fascistic in the sense that it’s real aim was to install a military dictator to benefit corporate interests. During the 30s, Hitler secretly became one of the wealthiest men in Germany.
Nazism was a distinct form of fascism because of the “leader” (furher) principle. One supreme leader would dispense with all of the wasteful disagreements and make the country great based on an almost divine will and vision. All the people would have to do was follow the cult worship of the supreme leader. (If this sounds a lot like Saddam Hussein, there is a reason for it.) While glib talkers like Herman Goering were able to make this grandiose, Nazism boils down to one super-narcissist exploiting a whole nation to the maximum extent for his (or her?) own whims. Such as world dominance. This being an aristocracy of one, it tends to be called right-wing.
During the 20s and 30s there were only half a dozen democracies in the world, and many people who studied power didn’t think that they would withstand the onslaught of either the communists or fascists, which people like Churchill believe was imminent. Churchill (half English, half American incidentally), like virtually all non-religious right Americans, was against both communism and fascism as they were assaults against liberty. One was an abuse of the power of the right, another was an abuse of the power of the left.
Democrats (I am one) are left-wing because they sit on the left side of the chamber in the metaphorical sense that they believe in one person, one vote. Rule of the people. Demos actually is Greek for the masses, and it is something of an insult by Aristotle, who didn’t particularly appreciate everybody having a vote, as not everybody had the time to sit around contemplate the issues. Athens was the “democracy” that Aristotle was criticizing.
Republicans (some of my closest relatives) are right-wing because the damn fools won’t do anything that isn’t good for business, as though that were a legitmate organizing principle. Oh no, wait. A Republic is a form of democracy by representation, much like the United States Senate. Each group is entitled to elect one or more representatives. But much like the US Senate, the representatives do not necessarily represent the same number of people, and in some instances, the underlying constituencies can be wildly disproportionate, like the US Senate. (But that is another ax to grind on another day.)