I can see that several people are looking for this (or maybe one person multiple times), so I hope this helps.
This discussion withing the Platonic corpus is in the Kaliopolis (a.k.a. Republic), in Book VIII, where Socrates explains the deterioration of government from the just city to tyranny. As far as the just city is concerned, it would take way too long to explain that and all the other government forms, but if people want to read my take on it, please comment on this post and request it and I will do my best to respond as quickly as possible.
So, what is a timocracy? Well, the etymology of the word is Greek (surprize, surprize), from the words timia which means “honor” and kratos which means “power.” In an nutshell, it means power of the honorable. It is basically a military state. The shift that has happened here from the just city is that it is no longer the philosopher, the reasonable and rational person who rules, but the spirited (willful) and courageous person. Socrates says that this is definitely worse than the just city, but not that bad, since one would have to be fairly intelligent in order to be a successful military leader (which means they would make good decisions concerning the people under them) and because honor is dependent on other people. This second reason can actually both raise the quality of the timocracy and lower it. Because honor is dependent on other people, one seeking to become the leader in a timocracy may do everything in their power to keep his people happy, which in itself could turn bad, but there is no concrete standards for judging who should be considered honorable and who shouldn’t. Let us take two figures from World War II, General Patton and General MacArthur. There are many people who love Patton for all of his achievements during WWII and for his “give ’em hell” attitude, but there is definitely a rather large group that think he was a drunkard and a deranged moron who won so many battles simply because no one else would be crazy enough to do what he did. MacArthur on the other hand, many people love him, think he was one of the greatest figures of US history, but Eisenhower (and he is not alone in this) did not have much love for him. At some point he said, “I studied dramatics under General MacArthur” (I knew APUSH would be useful some day). Patton or MacArthur, what are we to make of them, honorable or not? See the dilemma? This was further compounded if we did not live in our current day and age, where not only were attacks a whole lot more vile (at some point, after Cicero’s third marriage, he was accused to his face for having sex with his sixteen year-old daughter, which was not true, imagine what would happen if someone said that for Reed today), but people had far less neutral information, so polarization was a whole lot more likely.
Next we have oligarchy. Etymologically, this word comes from oligos, Greek for “little” or “few” and arche, Greek for lead. It basically means leadership by a few people. I have heard people say that the just city is an oligarchy, which if we go my Socrates’ definition, is most emphatically wrong. Socrates explains that in an oligarchy, the soul has further deteriorated into being governed by its desires, not the will, and not reason. Nonetheless, though desires occupy the highest position in the oligarchic soul, they are followed by reason, since it takes a lot of knowledge to be successful in business. This is the government where merchants flourish (think of Ancient Carthage or Phoenicia), since the whole mentality of the state is governed by the principles of business. All in all, Socrates says that even though the hierarchy of the soul has definitely gone wrong, this mode of government too is not horrible to live in, since, if you have some wisdom, you can probably get somewhere under this regime. Of course, the problem is that the people who have power would never willingly give it up, so there is an infinite race to stay on top. Also, this type of regime is not typically very courageous or very warlike in general, since war disrupts business (war when this particular state is involved of course, war between two foreign states is great for business).
As far as democracy is concerned, you can see my “Democracy and Freedom” post, which deals with the problem of democracy in great detail.
Last and most definitely least, comes tyranny. This is the regime where the whole of the human soul is overthrown. Socrates says that democracy is usually to end up into a tyranny, because in times of strife people will gravitate toward a strong figure (reminds you of anyone in the 1930’s? I’ll give you a hint, he was from Austria). The soul of the tyrant is solely governed by the desire to obtain more power, so the well-being of the people is, to him, only an afterthought, specifically if it helps him obtain more power. However, Socrates points out that not only are the people miserable in this regime, but the tyrant himself is the most miserable among them. Because he knows only one measure for everything, power, he thinks everyone is out to get him, so he himself is made a slave of fear. It is fear of people going for his head that causes him to be cruel toward the people under him. Of course, when people (either those under him or others) have had enough of his cruelty, they rise up and demand his head, which was what he was trying to prevent in the first place, which makes him more cruel, which makes more people rise up against him and, eventually, have him overthrown. His fear of losing power turns into the principal reason why he loses power, the whole thing is a self-fulfilling prophecy.
I realize that I have given just a blurb on the regimes, but if anyone wants me to treat any of them in more detail, please comment and I will be happy to help.