You can read the Statesman online in English at the Perseus Project or in French at Wikisource.
Since it picks up immediately after the Sophist, this dialogue has the same characters. In this dialogue, however, it is Theaetetus who is only present and never speaks, and Socrates the Younger who is the young man interacting with the Eleatic Stranger.
Socrates opens the dialogue by thanking Theodorus for introducing him to Theaetetus and the Eleatic Stranger, and Theodorus replies that soon he'll be indebted three times over when they have worked out the profiles of the statesman and of the philosopher as well as that of the sophist. Socrates jokes that he's disappointed that Theodorus, expert in arithmetic and geometry, would make such a simple mistake in math: he's adding the sophist, the statesman, and the philosopher as if they were of equal value instead of weighting them according to proportion. Theodorus likes the joke and says that he'll get even with Socrates later for it, but in the meantime he asks the Stranger to go on and discuss either the statesman or the philosopher, whichever he chooses to do first. The Stranger suggests that they give Theaetetus a rest and let Socrates the Younger take over the burden of interrogation for a while, and Socrates remarks that it's appropriate, since both young men are like Socrates in different ways, and Socrates the Younger will at some point answer questions from Socrates. And so the rest of the dialogue proceeds.
One peculiarity is that it's unclear which Socrates ends the dialogue: it could be either the older or the younger Socrates who remarks on the excellence of the Stranger's portrait of the statesman.
The Myth section of the dialogue is the most discussed (and also the most contentiously discussed) part of the dialogue. The Eleatic Stranger is discussing the sense in which the true king is shepherd of human beings, but they've run into the problem that there are so many look-alikes competing for the title. So to clear the field a bit, they need to go a different route. So the Stranger suggests they should bring in "an element of play" (268d) and look at stories. He notes three in particular: "the portent relating to the quarrel between Atreus and Thyestes" (268e), the golden age of Chronos (269a), and "the report that earlier men were born from the earth and were not reproduced from each other" (269b). He says that these are actually three different ways of talking about the same thing. (They might seem to be a weird mix, but they are all myths that turn out to be related to politics in various ways, and they are all stories in which a prior situation suddenly changes, i.e., in which there is a sharp difference between the way things were originally done and they way they are done later.) He puts them all together and makes a story in which the heavens went an opposite direction (the Atreus/Thyestes myth), people were originally born from earth because their lives were reversed and instead of ending up in the earth after death they began in the earth before they sprang from it as old men and women (the autochthony myth), and people were originally ruled by gods (the age of Chronos myth):
A god tended them, taking charge of them himself, just as now human beings, themselves living creatures, but different and more divine, pasture other kinds of living creatures more lowly than themselves; and given his tendance, they had no political constitutions, nor acquired wives and children, for all of them came back to life from the earth, remembering nothing of the past. While they lacked things of this sort, they had an abundance of fruits from trees and many other plants, which grew not through cultivation but because the earth sent them up of its own accord. For the most part they would feed outdoors, naked and without bedding; for the blend of the seasons was without painful extremes, and they had soft beds from abundant grass that sprang from the earth. (271e-272b)
However, at some point the "steersman" of the cosmos stopped turning it in its direction, and thus it began to turn the other way; then all the gods stopped directing the realms of the cosmos they directed, and after tumult and confusion, the cosmos (as a living thing) began to manage itself, remembering as best it could how it had originally been guided. But over time disharmony and discord added up; and things were done in the reverse direction. Just like the cosmos that we imitate, we currently live trying to manage ourselves under a reversed regime until the steersman rights the universe again, making do as best we can with the remembered gifts of the gods we received long ago (fire, crafts, domestication).
The point of the story is to point to a flaw in thinking of the statesman along the lines of a herdsman: it is effectively to give the statesman a role that can only exist in a Golden Age in which human beings are ruled by gods. The divine herdsman is far greater than any king, and statesman of our day are necessarily very much like the people they rule, not as different from them as a herdsman from the animals he herds.
Paradigm and Measure
The Stranger then reflects on the importance of models (paradeigmata):
It's a hard thing, my friend, to demonstrate any of the more important subjects without using models. It looks as if each of us knows everything in a kind of dreamlike way, and then again is ignorant of everything when as it were awake. (277d)
In the use of the model we are discovering how to make true judgments about something by comparing it with something else. And the model is necessary precisely to deal with the problem we are having with the statesman of distinguishing him from many things that look like him (as the Stranger notes, we are looking for men, but we keep getting with them centaurs and satyrs, i.e., men-like non-men). So we use the models to focus in on the statesman in the proper sense, eliminating the look-alikes. The Stranger chooses weaving as a model to get them farther than herding alone could.
The discussion of weaving leads to another key point: that of excess and deficiency. When measuring something, there are two kinds of measuring we could use. One kind is a relative measure, in the same way that we measure things by saying one thing is greater or smaller than something else. But another kind, and the kind that is relevant to any kind of art/skill/craft/expertise (techne), is that of measure (to metron). For every craft or skill, by its very nature, there is a way of distinguishing what is excessive and what is defective, and thus a way of identifying what is rightly done. If you are a weaver making good cloth, it is because you are finding this measure. Thus this measure distinguishes good and bad, beautiful and ugly, and it is based on the way in which things are generated or production. (This relates to the myths, incidentally, since all the myths were said to be about different ways things come to be; the divine herdsman is not an appropriate way of thinking of the statesman, because he existed in a different order of production or generation, when things were produced or generated in different way.) It is the due measure (to metrion), the decorous/fitting (to prepon), the timely (to kairon), the due (to deon).
This is all relevant to classification (which tries to divide things 'in the middle', that is in a way that is appropriate to the thing in question, failing not by an excess nor by a defect), and to statesmanship (which as a skill will be concerned with its own kind of due measure).
Typology of Regimes
The Stranger gives a very influential typology of regimes: we have monarchy (rule of one), and oligarchy (rule of a few), and democracy (rule of many). Monarchy is of two kinds, tyrannical and royal, and the difference has to do with things like force and consent; oligarchy is likewise of two kinds, oligarchical and aristocratic. Democracy, however, tends to go by the same name regardless. The best measure of the quality of a constitution is whether those who rule have the skill of statesmanship, but as it is often difficult to find people who have the requisite skill, the second best is that the written documents or ancestral customs of the society imitate, roughly and approximately, the kind of regime you would have under someone with this skill of statesmanship, despite sometimes not fitting situations perfectly. This is the real dividing point between tyranny and good kingship, and between oligarchy and aristocracy.
Because the existence of the statesman cannot be guaranteed, and because it can be hard to discover him even when he does, it is essential "for people to come together and write things down, chasing after the traces of the truest constitution" (301e). When you have monarchy with good written laws, it is the best kind of regime; but when the laws aren't good, it is the harshest. On the other hand, if a democracy with good laws is the least good of all good regimes; yet it has the corresponding feature that a democracy with bad laws is the least bad of all bad regimes.
Definition of Statesman
In leading up to the definition of the statesman, the Stranger suggests something "astonishing" (306b): the city needs temperance (sophrosyne) and fortitude (andreia), but that these are in some way opposed to each other. Now, of course, all 'parts of virtue', as the Stranger calls it go together, but the kinds of qualities that these parts of virtue are concerned with can at times oppose each other to the detriment of the whole. Thus, following the model of weaving, we get the statesman:
Then let us say that this marks the completion of the fabric which is the product of the art of statesmanship: the weaving together, with regular intertwining, of the dispositions of brave and moderate people--when the expertise belonging to the kind brings their life together in agreement and friendship and makes it common between them, completing the most magnificent and best of all fabrics and covering with it all the other inhabitants of cities, both slave and free; and holds them together with this twining and rules and directs without, so far as it belongs to a city to be happy, falling short of that in any respect. (311b-c).
* This dialogue in a sense has to deal with a classification problem opposite to that of the prior one. When the Stranger began dividing arts, the sophist ended up spread across several different divisions, and the problem was to make sense of why classification was so elusive. The statesman, on the other hand, turns out to be easy to locate -- but it turns out to be difficult to come up with any classification that fits only the statesman.
* One of the interesting things about both this and the prior dialogue is that in a sense they are examples of how to reason through figurative language -- Is the sophist better characterized in terms of hunting or selling? Is the statesman better characterized in terms of herding or weaving? I'm not sure that this is the intent; but almost the entire apparatus of the dialogues (division, paradigmatic cases, coordination of different 'guises', checking with likely stories) is well-suited for exploring metaphors. The reason in general, I think, is that figurative language is generally a form of indirect classification; so there is inevitably going to be some connection. It's simply brought out more clearly by the way the Stranger uses models. And this is perhaps a particularly appropriate conclusion if we think of these two dialogues as following Cratylus.
* The dialogue brings up the midwife's art again, in talking about herdsmen (268a-b). It's a passing mention, but probably a deliberate tie of some sort to Theaetetus, although I'm not sure how.
* Since the Statesman is an indictment dialogue, it is worth noting that it raises clearly and explicitly the issue of corrupting the youth.
Suppose anyone is found inquiring into steersmanship and seafaring, or health and truth in the doctor's art, in relation to winds and heat and cold, above and beyond the written rules, and making clever speculations of any kind in relation to such things. In the first place, one must not call him an expert doctor or an expert steersman, but a stargazer, some babbling sophist. The next provision will be that anyone who wishes from among those permitted to do so shall indict him and bring him before some court or other as corrupting other people younger than himself and inducing them to engage in the arts of the steersman and the doctor not in accordance with the laws, but instead by taking autonomous control of ships and patients. If he is found guilty of persuading anyone, whether young or old, contrary to the laws and the written rules, the most extreme penalties shall be imposed on him. For (so the law will say) there must be nothing wiser than the laws; no one is ignorant about what belongs to the art of the doctor, or about health, or what belongs to the art of the steersman, or seafaring, since it is possible for anyone who wishes to understand things that are written down and things established as ancestral customs. (299b-d)
Given that this is part of a very complicated arguments based on hypotheticals that are recognized that it is odd, it is difficult to know exactly what to make of this -- but it is notable that it is even brought up, particularly given that Crito has the Socrates representing the laws speaking on their own behalf. And it is noticeable that the Stranger makes quite clear that the laws, not being statesmen, cannot adapt to every situation, and therefore will not be right for every situation -- but will still need to be upheld.
There's obviously some set of implications here. But these implications are obviously indirect and would have to be teased out -- and, as noted before, there is no consensus about what these implications are. A difficulty with interpreting Plato -- sometimes the main argument is in what he is deliberately not saying.
Quotations are from C. J. Rowe's translation in Plato, Complete Works, Cooper and Hutchinson, eds., pp. 294-358.