The Seventh Letter is, without any question, the most important of the epistles. While its authenticity is still disputed, and no one earlier than Cicero (in the Tusculan Disputations) identifies it as Platonic, it is the one that is most actively defended as authentic, and there is a general consensus that if any of the Platonic Letters are authentic, the Seventh is. One line of argument for its authenticity is that (unlike some of the letters) its historical claims are fairly plausible (there isn't anything that could be anachronistic, for instance). Another is that its vocabulary is reasonably Platonic. But the single most influential argument is based on the philosophical content: whoever wrote this letter, if not Plato, had both a philosophical insight and a manner of expressing it worthy of Plato. Even people who regard it as inauthentic generally concede to it intelligence and a good Platonic style.
One of the arguments against its authenticity is that the story the letter gives of itself, of how it came to be written, is a bit convoluted and difficult to understand. The most important argument against its authenticity, however, is also content-based: the Seventh Letter says that there are Forms or Ideas of artifacts, whereas Aristotle tells us that Plato denied that there were Forms or Ideas of artifacts. Those who regard it as inauthentic are usually content to leave the author as unknown, although occasionally one finds a scholar who will venture that it might be by Speusippus, Plato's nephew and successor as head of the Academy.
It probably is the case, however, that whether a Plato scholar accepts it as authentic or not generally comes down to whether he or she wants it to be: it would provide a fascinating, if incomplete, insight into Plato's philosophy, but at the same time the interpretation of Plato that it suggests is not one people always find antecedently likely on other grounds, and many people would have to revise their preferred interpretations of Plato if it's taken to be authentic. However, it also seems to me that even people who think it is not authentic tend to be influenced by it at least indirectly, in their assumptions about the course of development of Plato's thought, since a lot of assumptions made about Plato himself have very little ground unless the Seventh Letter is more or less right in its account of Plato's early life.
The letter is an open letter addressed to friends and followers of Dion. Plato notes that they have written him, saying that since their aims are the same as Dion's, he should cooperate with them, and he replies that if their aims really are the same, he agrees, but only if they are the same. Plato knows Dion's aims personally, not by mere conjecture; he came to Syracuse at about forty years of age, when Dion was a young man, and it was at that time that Dion formed the major opinions that would last until his death: "the Syracusans, he though, ought to be free and live under the best of laws" (324b). He then gives an account of how such convictions arose, starting with his own case.
When he was young, Plato, like most young men, thought he would go into the active life of politics, but at about the time this was appropriate, the democratic government was overthrown and the oligarchical government of the Thirty Tyrants was established. Because he had relatives who were members and associates of the Thirty (he doesn't specify, but we know from other sources that his cousin Critias was a member of the Thirty itself, and his uncle Charmides was a member of the closely associated Ten who governed the Piraeus), he was invited to join their cause. He hoped that they would lead the city from injustice to justice; but he soon saw that the Thirty were so terrible that they made the unjust democratic government prior to them seem like a golden age; in addition, they treated his friend (philon andron) Socrates very poorly:
Among their deeds they named Socrates, an older friend of mine whom I should not hesitate to call the wisest and justest man of that time, as one of a group sent to arrest a certain citizen who was to be put to death illegally, planning thereby to make Socrates willy-nilly a party to their actions. But he refused, risking the utmost danger rather than be an associate in their impious deeds. (324e-325a)
(Compare Apology 32c.) Because of this he disassociated himself from them; and when they were overthrown and he considered returning to politics again, he was less enthusiastic. The restoration was turbulent and, of course, in the aftermath his companion (hetairon, often used for a co-worker or a partner in an enterprise or project) Socrates was tried and killed:
By some chance, however, certain powerful persons brought into court this same friend Socrates, preferring against him a most shameless accusation, and one which he, of all men, least deserved. For the prosecutors charged him with impiety, and the jury condemned and put to death the very man who, at the time when his accusers were themselves in misfortune and exile, had refused to have a part in the unjust arrest of one of their friends. (325c)
By this point Plato had a low opinion of politics in Athens generally, and it only got lower; but looking at other cities, he found that this seemed to be a general problem, not something exclusive to Athens. It is not easy to govern cities rightly without men who are friends and trusty companions (philon andron kai hetairon piston):
At last I came to the conclusion that all existing states are badly governed and the condition of their laws practically incurable, without some miraculous remedy and the assistance of fortune; and I was forced to say, in praise of true philosophy, that from her height alone was it possible to discern what the nature of justice is, either in the state or in the individual, and that the ills of the human race would never end until either those who are sincerely and truly lovers of wisdom come into political power, or the rulers of our cities, by the grace of God, learn true philosophy. (326a-b)
(Cp. Republic Book V, 473d.) Mulling this over, he toured the Greek colonies in Italy and Sicily. He was not impressed, thinking their style of life suited only to produce intemperance and excess, and that style of life a guarantee of corruption and restless change in government. At this point he met Dion, whom he found a quick student, and Dion began to develop the opinion previously noted, that the only appropriate constitution is one in which one has a just government under equal laws. He hoped that Dionysius the Elder might come to share his opinions, and to this end convinced Dionysius to invite Plato to Syracuse. Plato was reluctant, well knowing that young people get sudden enthusiasms and then change their minds as quickly, but he went, finally considering that if there was any time to try to make a properly philosophical view of politics take real form, now was the time to take courage and do it. When he arrived at Syracuse, however, the court was full of nasty rumor and innuendo about Dion, whom Plato defended. Only a few months later, Dion was driven out; Dion's friends were afraid for their lives at that point, but as it happens Dionysius himself was worried that Dion's friends might be the cause of something drastic if he did not conciliate them. But Plato was practically a prisoner under the guise of Dionysius's friendship, for quite a while.
Thus Plato's first trip to Syracuse. At this point in the letter, Plato considers the topic of advice, which turns out to be directly relevant to the question of revolution. If you are advising a sick man on the way to get well, first you tell him to change his life to a more healthy way of living. If he does this, then you can move on. If he refuses, however, there's very little one can do. The same thing happens when we are dealing with cities. If the city is already well run, it's reasonable to set about advising the citizens on how to contribute to what the government recognizes as good. If the city has a corrupt government, however, the wise man will refuse to advise people on how to contribute to what the government recognizes as good; he will advise the city to change, and if it does not, there is not much else to be done. One could try to compel the one being given advice -- in the case of a government, this would be a revolution -- but this is impious, like compelling one's parents to follow one's advice. "If what he thinks is best can only be accomplished by the exile and slaughter of men, let him keep his peace and pray for the welfare of himself and his city" (331d).
Therefore Plato advises them, as he and Dion used to advise Dionysius, to develop mastery over themselves and to win trusty friends and companions (pistous philous te kai hetairous). This was the mistake of Dionysius the Elder: "he was poor in friends and loyal followers (andron philon kai piston), and the possession or lack of these is the best indication of a man's virtue or vice" (332c). Dion and Plato advised Dionysius to "induce others among his relatives and companions to become friends and partners in the pursuit of virtue; but above all to become a friend to himself" (332d). Dion eventually returned with an army and taught Dionysius a lesson the hard way. They Syracusans then believed lies about Dion, and Athenian comrades he had gained rose up to help kill him; but, Plato notes, there was an Athenian (himself) who would not betray Dion even when honors were to be had for doing it; he was bound to Dion by free and common cultivation (eleutheras paideias koinonian), which is a stronger bond than mere kinship in soul and body like casual friends have.
Thus, says Plato, the course is clear: "Do not subject Sicily nor any other state to the despotism of men, but to the rule of laws" (334c). Despotic power is always destructive, and it would be better to die honorably and nobly than to live defiled by wickedness. Those who do the latter are worse off both in this life and in the next. What we actually have with Dionysius, and with the murder of Dion, are injustices against humanity:
If in his [Dionysius's empire there had been brought about a real union of philosophy and power, it would have been an illustrious example to both Greeks and barbarians, and all mankind would have been convinced of the truth that no city nor individual can be happy except by living in company with wisdom under the guidance of justice, either from personal achievement of these virtues or from a right training and education received under God-fearing rulers....[If Dion had lived and fulfilled his plans to resettle Sicily and equalize the laws,] such deeds accomplished by a man of justice and courage and temperance and philosophy would have produced in the multitude the same respect for virtue which, if Dionysius had listened to me, would have made its saving appearance, one may say, among all mankind. (335d, 336a-b)
But it was not to be. Plato encourages the friends of Dion to imitate him and try to finish what he could not; get help to resettle Sicily and give the Syracusans equal laws, i.e., laws for common good and not private good.
Plato then continues his narrative of the relation between himself and Syracuse by relating his last trip to Syracuse. Dionysius had extracted a promise from him previously that he would return, and Plato agreed on condition that Dion also could; Dionysius never fulfilled the condition, but he did ask Plato to return allegedly because he was interested in philosophy again. Plato didn't rate this as worth much, but he was encouraged by Dion and others (like Archytas, the addressee of other letters) to return, and so he did, hearing all the while of the wonderful progress Dionysius was making in philosophy and that Plato's return was important for making sure that Dionysius continued his friendship with Archytas and others. He set out to test what Dionysius really knew of philosophy: he explained just how much learning, and effort, and time, and difficulty, was required to do philosophy, giving them a glimpse of the sheer sublimity of that to which it tends. True lovers of wisdom are not turned off by such things; it fires them up, even more eager to meet the challenge. Others, like manual laborers, will hear and conclude that it's just not the life for them. But another group of people will hear and conclude that they don't actually need to make the effort; they have already achieved everything important. Dionysius fell into this third group; he claimed to have knowledge of many of the most important things; he even went on to write a book about the things he and Plato talked about, conveniently not citing Plato anywhere.
This launches Plato into what is often known as the Philosophical Digression, but it is not a digression, since he is showing just how absurd Dionysius's attitude was (and, it should be said, the Seventh Letter, even in narrating events has been entirely philosophical up to this point as well). We have the famous comments about writing (cp. Phaedrus) and teaching:
There is no writing of mine about these matters, nor will there ever be one. For this knowledge is not something that can be put into words like other sciences; but after long-continued intercourse between teacher and pupil, in joint pursuit of the subject, suddenly, like light flashing forth when a fire is kindled, it is born in the soul and straightway nourishes itself. (341c)
He explains this using the doctrine of the Forms or Ideas. This argument could fill an entire post; it mostly just needs to be read in its own context. But the essential idea is that Plato is doing to the reader what he was doing with Dionysius: he is giving us the test, and emphasizing the importance of teacher and student working together to achieve the goal. He concludes:
...whenever we see a book, whether the laws of a legislator or a composition on any other subject, we can be sure that if the author is really serious, this book does not contain his best thoughts; they are stored away with the fairest of his possessions. (344c)
Thus Plato saw entirely through Dionysius; there was no true philosophy there, but only an extraordinary arrogance masquerading as love of wisdom. He goes on to describe some of Dionysius's related failings, and how he barely managed to get out of Syracuse this time, with the help of Archytas. When Plato told Dion, he was ready for vengeance, but Plato tells him that he is with him only if he desires friendship and the accomplishing of good things.
Thus we return again to Dion's aim: "the best and most just constitution and system of laws" (351c). He merely erred in not realizing the sheer extent to which wicked people will go.
There is so much in the Seventh Letter that one could talk about it for a very long time. But I want to highlight one thing: it is a letter about the centrality of virtuous friendship to the health of a city. Friendship, virtue, philosophy, and rule of law are all interlinked; only through genuine and trustworthy friendships can we achieve the philosophy and cultivate the virtue that makes it possible to aim at the true common good of everyone, which is necessary to equalize the laws and build a city in which laws rule, not despotic men.
The Eighth Letter seems mostly to get by on the coattails of the Seventh, because they have a number of similarities. The most common complaints raised by those who judge it inauthentic is that it makes some historical claims that are difficult to square with other historical sources we have; but some of this might simply be compression.
The letter is addressed to the relatives and friends of Dion, like the Seventh, and gives advice on how to quell the tumult that Sicily is undergoing in the struggles between Dionysius and the friends of Dion. Plato's suggestion for a compromise between the tyrannical party of Dionysius and the democratic party of the friends of Dion is a kingship governed by laws, giving the example of the ancient legislator Lycurgus, who arranged things so that "law became the lord and king of men, not men tyrants over the laws" (354b-c), and warning the democratic party that an untimely thirst for freedom can result in destructive anarchy. Obedience and liberty are both good in moderation, but destructive in excess. But this does require the rule of law: "Due measure is found in obedience to God, the absence of measure in obedience to men. And the god of wise men is the law; of foolish men, pleasure" (354e). He then imagines what Dion might say to Syracuse:
Of the three goods--soul, body, and wealth--your laws must give the highest honor to the excellence of the soul, the second place to that of the body, and the third and lowest rank to wealth, since it serves both body and soul. (355b)
The imagined Dion continues by proposing three kings, dividing up the power among them according to law, and also giving power to a body of guardians (who handle matters of war and peace, and also capital crimes) and an assembly.
Quotations from the Platonic epistles are all from Glenn R. Morrow's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 1634-1676.