Plato, in his Republic, describes several regimes, starting with his ideal regime, the rule of philosopher kings, called aristocracy, or rule by the best, in the book. In this regime, the philosophers are forced to take time out from philosophizing to rule, and then pass the burden on as soon as they can to get back to interesting things, like contemplation. Below the philosophers are guardians, or warriors, and below them are the workers. The philosophers and warriors live a totally communal life, with not only property but women and children in common, in order to prevent them from having a conflict between public and private interest. It's not made clear whether the ordinary citizens would also live a communal life, but Aristotle (who studied under Plato) said they would, in order to keep the rulers from being envious.
Moderns tend to divide the mind (or soul; the Greek psyche means both) into the id, ego, and superego. (See Sigmund Freud.) The Greeks divided the soul into three parts of their own, with only one part in common. The Greek parts were the desiring part (which is like what we call the id, but without so much implication of suppressed deviant sexuality), the spirited part, and the reasoning part.
The social classes in Plato's ideal city correspond to the parts of the soul; the philosophers to the reasoning part, the guardians to the spirited part, and everyone else to the desiring part. His ideal city represents the idea soul; reason should rule spiritedness and together they should rule desire. His main point is about how the soul should be ordered. There's been some debate over whether he actually meant his aristocratic regime to exist in practice and whether he even thought it could exist in practice. If it ever did, it would strike us as totalitarian, resembling Fascism or the worst forms of leftism. But moving on, his other regimes also match types of souls. (All the souls are ruling class souls, as we'll see when we get to democracy.)
If the ideal state respresents rule by reason, the highest part of the soul, the regimes decline as they get further away from reason and more toward desire. Next best after aristocracy is timocracy, where the spirited part of the soul prevails. The timocrat is the lover of victory, and his regime is a military one like Sparta (which Plato did not mention; he was a citizen of Sparta's enemy Athens) or the cities of Crete. After that, the descent goes from the spirited part to the desiring part, with the oligarchic regime, a regime ruled by misers and rich men, like Medieval Venice. The oligarchic soul is devoted to necessary pleasures, such as food, which leads an oligarchic man to save up money in case of need. Descending from necessary to unnecessary pleasures, we come to democracy. The democratic man is not a commoner, but a member of the elite who is easy-going, devoted to his harmless pleasures; a playboy. The pleasures here a still lawful, but when they break out into unlawful pleases the descent has reached the bottom, tyranny.
Democracy is unique for including men of every type; everyone is his own regime. Thus we see aristocratic men, like Socrates, talking to tyrannic men (who might later try to be tyrannic in the usual sense and thereby cause political problems).
Thus we get:
(What is called "republic" above is his "polity".)
(This is very incomplete. Someone should add some stuff about mixed regimes and about modern theories, not to mention practice.)