Timaeus as we have it seems to have been intended as the first in a projected trilogy of dialogues, to be followed by Critias and a dialogue devoted to Hermocrates. However, the third does not seem ever to have been written -- even in antiquity no one knew it -- and Critias as we have it is incomplete, and, indeed, is usually thought never to have been completed. Timaeus is one of six dialogues that Aristotle both attributes to Plato and designates by its title, and he refers to it on multiple occasions, so it has the strongest possible external attestation.
Timaeus can be read in English at the Perseus Project and in French at Wikisource. (The Timaeus is a fairly long dialogue, and not wholly easy reading; but if you're reading along, I strongly recommend you at least read the opening, from 17a to 31b. The opening frame applies to the much shorter and easier Critias as well, and the tail-end of that section will also give you a taste of Timaeus's discourse. But there is no dialogue in the Platonic canon that will explode your mind with crazy images than the Timaeus will; it's worth reading as much of it as you can.)
(in order of appearance)
Socrates opens the dialogue, but he is, beyond some early comments, not a major contributor to the discussion, and mostly spends his time listening to Timaeus talk at great length.
Timaeus of Locri
Timaeus is the primary speaker in this dialogue. He is mentioned in Critias, but is unknown outside of Plato, and it is a matter of considerable dispute whether he is historical or fictional.
Hermocrates of Syracuse
Hermocrates is mentioned by Thucydides and by Xenophon in the Hellenica. He was originally a general for Syracuse, although he was removed because of his lack of success in battle; however, he became advisor to the general Gylippus, and, having what seems to have been a good diplomatic sense to make up for his military haplessness, he may have been a major contributor to Syracuse's building of the alliance with which they resoundingly trounced the imperialist Athenians.
The problem we have here is that there are two Critias's. The most obvious one, who is found as a character elsewhere in Plato's dialogues, is the oligarch who became the most important of the Thirty Tyrants. This is the assumption that has usually been made by commentators through the ages. However, John Burnet in 1914 argued very firmly that this was not the case: the Critias here is the grandson of that Critias (it was very common for Greek men to have the same name as one of their grandfathers). That would make this Critias Plato's own great-grandfather. There is a still a minority of scholars who will argue that the oligarch is intended here, but most take it to be the older Critias, because the indications of time (e.g., in Critias's account of the transmission of the Atlantis story) fit better. This makes sense to me, although Plato is not afraid to distort dates when it meets his purposes.
Socrates opens the dialogue by asking Timaeus where the fourth guest is. (The fourth guest is never named and never described; some have speculated that if the trilogy had been completed we would have learned who he was.) The three guests who are presents will be returning Socrates' hospitality by speeches, because the day before Socrates had held them fascinated recounting his own speech. Socrates then goes on to summarize the ideas he had discussed, and his account basically runs through a significant portion of the argument of the Republic, describing the ideal city that gives that dialogue its name. Socrates continues, however, by suggesting that he is somewhat frustrated because it is like seeing a magnificent animal in a painting and wanting to see it in motion. So he says that he'd love to hear a speech showing the city in action: how it goes to war, how it deals with other cities, and the like. So the three guests are to give their own speeches reflecting on this aspect of the city; all three are excellently suited because they are familiar with philosophy and politics alike.
Hermocrates remarks that they are all excited about it, and that Critias had mentioned a story that might fit the bill perfectly. Critias says the story is "a very strange one, but even so, every word of it is true" (20d). Critias got it from his grandfather (also called Critias), who got it from his friend Dropsides, who got it from the sage Solon, who got it from an Egyptian priest, who learned it as part of an Egyptian tradition tracing back eight thousand years, which recorded an event that happened a thousand years before that. In those days Athens was a great and flourishing city that had been established by Athena and Hephaestus:
And, being a lover of both war and wisdom, the goddess chose the region that was likely to bring forth men most like herself, and founded it first. And so you came to live there, and to observe laws such as these. In fact your laws improved even more, so that you came to surpass all other peoples in every excellence, as could be expected from those whose begetting and nurture were divine. (24d)
But great as Ancient Athens was, it was surpassed by another power, the Island of Atlantis, which ruled the Atlantic Ocean and increasingly came to dominate the Mediterranean, spreading its empire as far as Italy in Europe and as far as Egypt in Africa. Only Athens and Egypt stood in the way of its enslaving all of mankind, and it gathered its power to destroy them both. But Athens rose up, although forced to fight alone, and pushed back the armies of Atlantis, freeing all the other Mediterranean peoples. But it was a victory at great cost:
Some time later excessively violent earthquakes and floods occurred, and after the onset of an unbearable day and a night, your entire warrior force sank below the earth all at once, and the Isle of Atlantis likewise sank below the sea and disappeared. (25c-d).
Critias remarks that he was amazed to hear Socrates' description of the ideal city, because it fit so perfectly with what Solon had said. So what will happen is that Timaeus will start their discourse out by telling how human beings originated. Critias will then combine Timaeus's account of human nature with Socrates' account of ideal education to make up the citizens of the Ancient Athens of Solon's story.
Socrates then asks Timaeus to invoke the gods, which he does, and then begins his discourse, reasoning backwards to first principles. He insists, however, that we should be cautious in how we understand what he says:
Don't be surprised then, Socrates, if it turns out repeatedly that we won't be able to produce accounts on a great many subjects -- on gods or the coming to be of the universe -- that are completely and perfectly consistent and accurate. Indeed, if we can come up with accounts no less likely than any, we ought to be content, keeping mind that both I, the speaker, and you, the judges, are only human. So we should accept the likely tale on these matters. It behooves us not to look for anything beyond this. (29c-d)
Socrates agrees, and Timaeus continues his discourse, which takes up the rest of the dialogue. Timaeus ends on a vivid flourish:
And so now we may say that our account of the universe has reached its conclusion, This world of ours has received and teems with living things, mortal and immortal. A visible living thing containing visible ones, perceptible god, image of the intelligible Living Thing, its grandness, beauty and perfection are unexcelled. Our one universe, indeed the only one of its kind, has come to be. (92b)
* Socrates' account of his speech the prior day is very definitely the argument of a significant portion of the Republic (Books 2 through 5, in particular), but it is not the discussion we actually get in the Republic, which occurs at a different time of year with completely different people. The usual way of handling this is to say that Socrates simply re-told the discussion he had had at the Bendideia festival, and which is recorded in the Republic. Whatever the solution, Timaeus and Critias are explicitly linked to the Republic.
* Solon was one of the seven sages of Greece. Solon instituted a number of legislative reforms in Athens and then went traveling the world, supposedly so that the Athenians would not be able to pressure him to repeal any of his laws. He is also the earliest extant Athenian poet.
* The festival being celebrated is usually thought to be the Panathenaia (which is usually thought to be the subject of the famous Parthenon frieze), but the dialogue does not get more specific than saying it was dedicated to Athena.
The Apaturia is another festival mentioned; it was an especially important festival for boys.
* I will discuss Atlantis a bit more when I get to Critias, but it is worth noting here that Ancient Athens as described in the story shares features of both Athens and Sparta, and that there are features of the story that are reminiscent of (but more extreme than) the Battle of Marathon, when Athens held off a much more powerful Persian army.
* Notice that the whole point of Timaeus's discourse is to tell us about human nature.