Theaetetus is an aporetic or perplexed-conclusion dialogue; it is concerned with the definition of knowledge, and is very tightly argued. The dialogue is the source of a number of influential ideas, including that idea that we should become like God, the idea that philosophy begins with wondering, and the portrayal of Socratic method as midwifery. You can read an English translation at the Perseus Project and a French translation at Wikisource.
In the frame narrative:
Euclides of Megara
Euclides was a student of Socrates who founded the Megarian school of Hellenistic philosophy; most of our information about this school is either indirect or mere rumor, but there is good reason to believe that they had a significant influence on Stoic logic. Of Euclides we also have mostly indirect information or mere rumor, but Plato indicates he is also present in Phaedo, and he wrote Socratic dialogues, although none of them have survived.
Terpsion of Megara
We know next to nothing about Terpsion except that he is a friend of Euclides and, like Euclides, is present in Phaedo as well as this dialogue.
There is also an unnamed slave of Euclides
In the main dialogue:
Theodorus of Cyrene
Theodorus was a geometer and is mentioned in Xenophon's Memorabilia as such. According to at least one tradition, Plato studied mathematics under Theodorus.
Theaetetus of Sunium
Theaetetus would become a mathematician, and a number of important mathematical discoveries are attributed to him (although many are likely apocryphal). However, as suggested by this dialogue, he only lives about seven or so years after Socrates' death, dying before reaching the age of thirty.
Socrates of Athens (Socrates the Younger)
'Socrates' was a common name, so it is unsurprising that there are other people in Plato and Xenophon besides the Socrates, and this is one. We know almost nothing about him, but Aristotle seems to mention him once in the Metaphysics. While present, he doesn't actually talk in this dialogue.
There are also a number of unnamed young men.
Euclides opens the dialogue by asking if Terpsion just got in; and Terpsion says he has been there a while, looking for Euclides. Euclides was at the harbor and happened to meet Theaetetus, wounded and sick. He walked with him a while, and, on his way back home, started thinking back about a discussion he had with Socrates not long before his death. In that discussion, Socrates related another discussion he (Socrates) had had with Theaetetus. Euclides was struck by it at the time, so he made notes on it when he got home, and whenever he visited Socrates (presumably while Socrates was waiting for his execution), he would ask questions about it, so he could correct his impression. He then wrote it up as a dialogue. Euclides has one of his slaves read this dialogue to Euclides and Terpsion. (Thus we have a rather complicated structure here. Plato is giving us a dialogue in which a slave is reading a dialogue written by Euclides on the basis of notes he made from when Socrates had talked about a prior discussion between Theaetetus and Socrates.)
The internal dialogue begins with Socrates asking Theodorus if he has any promising students here in Athens; Theodorus mentions Theaetetus, who looks a little like Socrates. Socrates and Theaetetus begin talking about knowledge. This leads eventually to a discussion of Protagoras, much of which takes place between Socrates and Theodorus. This eventually transitions back to Theaetetus, and they discover that all three of Theaetetus proposals for knowledge are empty. Socrates ends the dialogue by saying that he has to go to the stoa of the king archonto meet the indictment brought by Meletus; but he insists that they all pick up the discussion again the next day.
It would be difficult to go into detail for all the arguments in this work, so to prevent ourselves from going on forever, we need to look at the argument in its broad sweeps. Several key points are worth mentioning.
One of the significant passages in this dialogue is the extended discussion of Socrates' art as maieutic or midwifery (149a-151e):
The difference is that I attend men and not women, and that I watch over the labor of their souls, not of their bodies. And the most important thing about my art is the ability to apply all possible tests to the offspring, to determine whether the young mind is being delivered of a phantom, that is, an error, or a fertile truth. For one thing which I have in common with the ordinary midwives is that I am barren of wisdom. The common reproach against me is that I am always asking questions of other people but never express my own views about anything, because there is no wisdom in me; and that is true enough. And the reason of it is this, that God compels me to attend the travail of others, but has forbidden me to procreate. So that I am not in any sense a wise man; I cannot claim as the child of my own soul any discovery worthy of the name of wisdom. But with those who associate with me it is different. at first some of them may give the impression of being ignorant and stupid; but as time goes on and our association continues, all whom God permits are seen to make progress--a progress which is amazing both to other people and to ourselves. (150b-d)
Socrates thus proposes to help Theaetetus through his own labors to give birth to "a multitude of beautiful things" (150d). But he notes that those who started out with him but thought that they were doing all the work have fallen in with bad company, unskilled in maieutic, so that they miscarried everything else they could have given birth to, and even lost the children they had had through the help of Socrates. (Socrates gives as an example Aristides son of Lysimachus, who is in Laches.)
One of the requirements of working with Socrates, however, is that you have to be willing to discover that when you seemed to give birth, you really only had a 'phantom' or a 'wind-egg'. This will happen, of course, as the dialogue continues, since all the proposed definitions of knowledge will fail. But even this prepares Theaetetus for future birth, or, if not, for being a better person anyway:
And so, Theaetetus, if ever in the future you should attempt to conceive or should succeed in conceiving other theories, they will be better ones as the result of this inquiry. And if remain barren, your companions will find you gentler and less tiresome; you will be modest and not think you know what you don't know. This is all my art can achieve--nothing more. I do not know any o fhte things that other men know--the great and inspired men of today and yesterday. But this art of midwifery my mother and I had allotted to us by God; she to deliver women, i to deliver men that are young and generous of spirit, all that have any beauty. (210c)
Protagoras, Heraclitus, Parmenides
Protagoras, as Plato understands him, takes knowledge to be simply a matter of perception or how things appear. Much of the dialogue is devoted to arguing against this Protagorean position. However, Socrates argues that this Protagorean position is also closely linked with a particular view, associated with Heraclitus, that everything 'flows' or continually changes. Thus there is a major opposition is in the background thruoghout the dialogue: the fluent philosophers, who like Heraclitus hold that everything changes, and the Parmenidean view that what is and is known, in a proper sense, does not change. One of the important points about how Socrates approaches this is that he takes the Parmenidean view to be clearly a minority position. Most of the poets, philosophers, sophists, hold the Heraclitean view, or, at least, like Protagoras hold positions that seem to require it.
This can be important because if I ask what something is, the Heraclitean view indicates that there is no definite objective answer -- if I gave a stable, steady answer, it is only by convention (X is knowledge, for instance, just because we call X-like things 'knowledge', regardless of how they change). It's worth pointing out that this directly links up with the question of the indictment -- if charges like impiety are purely conventional, then they are expressions of 'might makes right'; only if they can be talking about something objective, natural, can they be a matter of justice in any sense that does not equate justice with the will of the powerful.
There are two different ways of reading Theaetetus that are related to this opposition. Under one reading, Plato is focused on refutation of the Heraclitean position; on the other, he is recognizing serious problems with it but not necessarily taking it to be refuted outright. There is no consensus among scholars about which is the better reading.
(1) Knowledge is nothing other than perception. This is the view that Socrates attributes to Protagoras, and how he understands the Protagorean claim, "Man is the measure of all things." This ends up having a number of problems, including that it becomes difficult to see how one can distinguish true from false perceptions -- all perceptions are perceptions, so if knowledge is nothing other than perception, everyone knows exactly what they perceive. All perceptions are true; no one is wiser than anyone else, so that even the gods are not wiser than men; we have knowledge whenever we perceive something; anything we can say about perception should apply to knowledge, as well. There are problems with all of these implications. A solid Protagorean, of course, will have answers to these objections -- but we run into the problem of knowing that these responses are true, and with the apparent fact that these responses seem to require us to say that there is something known that goes beyond mere perception.
(2) Knowledge just is true judgment. Much of Socrates' response to this focuses on false judgment, because if knowledge is to be true judgment, there must be some definite way to distinguish true judgment from false judgment that is presupposed by the account of knowledge. Socrates argues at some length that the distinction cannot arise from sensory experience alone; we repeatedly get puzzles about what false judgments would be. And these puzzles have the ramification that there must be some particular element that is being left out by identifying knowledge with true judgment. We see this again in the case of juries: a jury can have true judgment without knowledge, because they are making use of limited evidence and getting most of it secondhand.
(3) Knowledge is true judgment with explanation (logos). This seems more promising, but Socrates argues that it fails, also; either 'explanation' just ends up meaning 'whatever we-know-not-what that makes true judgment knowledge' or it ends up falling short of knowledge in some way.
Thus all three definitions of knowledge proposed by Theaetetus fail -- they are 'phantoms' or 'wind-eggs'. But Theaetetus -- and the reader --- is better off for this discovery.
* Note that we have two impending deaths in this dialogue: that of Socrates, just before the process that begins the process toward his death, and that of Theaetetus, from war-wounds and dysentery. Euclides bridges the two.
* This is the only dialogue in which Socrates describes his art as midwifery; but his doing so contains an implicit answer to both parts of the indictment: he is not impious, but following the course laid out for him by the God, in accordance with his divine sign; and he is not corrupting the youth, but bettering them -- they instead are corrupted when they set out on their own or fall in with a bad crowd.
* Berkeley in his work Siris repeatedly refers to Plato, and the Theaetetus is explicitly referred to seven times. One of those times he provides an interesting comment on the two opposing camps:
348. Socrates, in the Theætetus of Plato, speaketh of two parties of philosophers...the flowing philosophers who held all things to be in a perpetual flux, always generating and never existing; and those others who maintained the universe to be fixed and immoveable. The difference seems to have been this, that Heraclitus, Protagoras, Empedocles, and in general those of the former sect, considered things sensible and natural; whereas Parmenides and his party considered to pan, not as the sensible but as the intelligible world, abstracted from all sensible things.
349. In effect if we mean by things the sensible objects, these, it is evident, are always flowing; but if we mean things purely intelligible, then we may lay on the other hand, with equal truth, that they are immoveable and unchangeable. So that those, who thought the whole or to pan to be...a fixed or permanent one, seem to have understood the whole of real beings, which, in their sense, was only the intellectual world, not allowing reality of being to things not permanent.
* All of this seems to have some kind of relation to Socrates' indictment and trial, but Plato doesn't bring it out explicitly. The following passage, however, provides what I suspect is an important clue in this regard:
Socrates: Then consider political questions. Some of these are questions of what may or may not fittingly be done,of just and unjust, of what is sanctioned by religion and what is not; and here the theory may be prepared to maintain that whatever view a city takes on these matters and establishes as its law or convention, is truth and fact for that city....It is in those other questions I am talking about -- just and unjust, religious and unreligious -- that men are ready to insist that no one of these things has by nature any being of its own; in respect of these, they say, what seems to people collectively to be so is true, at the time when it seems that way and for just as long as it so seems. And even those who are not prepared to go all the way with Protagoras take some such view of wisdom. (172a-b)
Thus Protagorean epistemology and Heraclitean metaphysics aren't just abstract concerns; having to do with what justice is, and what piety or impiety is, whether they are things we just make up as we please or not, they are quite literally matters of life or death for Socrates.
* One thing that occurred to me well after writing this is that the dialogue provides a clue as to what knowledge is, without going out of its way to signal it, and without telling us exactly what to make of it: the single most important insight into the nature of knowledge in this dialogue is not any of the three 'wind-eggs', but the Socratic maieutic itself, which we see both described and shown. Socratic maieutic is not knowledge, as the dialogue carefully notes -- but it also seems clear that it is a clue to what knowledge would have to be.
Quotations are from Myles Burnyeat's revision of M. J. Levett's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 157-234.