Here's the way I see the present economic spectrum
Communist:
There hasn't ever been a communist with national political creds in the United States. The nearest is probably the Wobblys (IWW) of the John Reed era or Longshore chief Harry Bridges. There have been numerous small communes that were somewhat successful over the years, but they are imbeded in the larger culture. By Communist, I mean that most or all of the means of production is owned and controlled by "the people", meaning the state. Whether this state is democratic, authoritarian or something else is a separate issue, although the large ones have all been authoritarian. Significant free enterprise is not allowed although there may be small businesses with government licenses.
Socialist:
There have been a few socialist politicians with national credibility. The most important was probably Eugene Debs. Most of the present national people who call themselves socialists really are Democratic Socialists. By Socialist, I mean that a sizable share of the means of production is owned by the state, but significant free enterprise is allowed although with regulation. Again, whether the state is democratic, authoritarian or something else is a separate issue.
Democratic Socialist:
Most successful countries over the last century have been Democratic Socialists. Major US leadership figures include Franklin Roosevelt, Bernie Sanders, Elizabeth Warren, and arguably, Abraham Lincoln. By Democratic Socialism, I mean that the majority of the means of production is free enterprise, but a significant amount are operated by the government or at least strongly regulated. Usually these are those industries that have been in some sort of market failure. The associate government is democratic or republican.
Capitalist:
Most successful countries have passed through a capitalist phase. By capitalist, I mean that the vast majority of the means of production is individually owned, there is little regulation, and there are few industries in market failure. It is inherently an unstable circumstance: businesses will try to grow and dominate markets, and without limits, some will succeed and drive the others out. It also tends to breed various labor abuses, including chattel slavery, wage slavery, debt slavery. The capitalist era in the US ended with the rise of "the guilded age" after the civil war. Most "businesses" were family farms and most industries were small and isolated enough that there was no monopoly. Interestingly some of the most successful were substantially sponsored by the government, e.g. the Springfield Armory.
Those that profited most, and those who see only the opportunity to maximize profit and not the collateral damage of capitalism and the unique circumstances that make it possible are always trying to "revert" to capitalism. Capitalism is highly transitional and can only thrive while there are literally hundreds of businesses in each market: enough to allow true competition and natural selection.
Oligarchist:
What happens when a capitalist society fails to deal with market failure. Individual businesses grow so large and powerful that they dominate their markets--often with tacit or explicit collusion from their competitors. Before long they can afford to buy any regulation they may face. Sometimes growth is for the good, but more often, the giant prioritizes profit over providing good service or decent wages.
Laissez Faire:
There has never been a successful Laissez Faire society. There are two failure modes: Chile and several others gave it a try but it went so badly they needed to put down near constant rebellions with extreme force, including secret disappearances. The other way tends to last longer: it's called A Dark Age. The tools of civil society break down--banditry is rampant and there is little or no distinction between bandits and police. There is no rule of law. There are people who claim to advocate Laissez Faire. They're profoundly ignorant.
There's also a political axis. How leaders are selected and replaced, what powers they have, how they respond to popular opinion, etc.
Authoritarian:
Leaders grab power by force, are free to do anything they want including manipulate the selection process. They invariably are subject to frequent coups d'etat, especially after a previous leader dies. There is usually extensive manipulation of media to quell almost constant grumbling from the powerless. How the economy works, whether there is rule of law or not, etc., are separate issues.
Aristocracy:
Leaders initially gain power by force but agree that replacement will be based on some predictable scheme, usually hereditary. This tends to reduce the violence after a leader has died. Again, the workings of the economy is separate. Monarchy is a different name for the same thing.
Republic:
Representatives gather and make decisions about what to do. How the representatives are selected is based on various mechanisms--sometimes they're elected, sometimes they are land owners, Ancient Athens drew lots to decide who would lead, various other systems have been used.
Constitutional Monarchy:
The same as a Republic but there is an aristocratic figurehead who may or may not have some power.
Democracy:
The people vote on most things. There have been no successful democracies larger than a few hundred voters. All have either regressed in some way or have functionally been a republic.
Fascism is Authoritarian Oligarchy: a dominant leader grabs and maintains power through alliance with monopolist or near monopolist businesses. There are some stylistic things that usually accompany fascism: persecuted people who are supposedly the cause of all of these problems, state control of an extremely dishonest media, etc. Technically, there are other ways to be Authoritarian Oligarchist, but fascism is the most common.
Chinese and Soviet Communism is Authoritarian Communism or Socialism.
There are several democratic socialist monarchies in western Europe: Holland, Sweden, etc. Politically, these are republics but they retain a figurehead. For example, functionally the German and Swedish political systems are pretty similar, but Sweden has a ceremonial king. Saudi Arabia is an oligarchic aristocracy. The US has regressed to its second oligarchic phase, having advanced to democratic socialism for most of the 20th century. The current president has very, very strong authoritarian tendencies and we may go all the way to Fascism if we're not careful.
