(1) Plato completed the dialogue, but most of it was lost relatively early.
(2) Plato intended to complete the dialogue, but could not for some unknown reason.
(3) Plato deliberately wrote the dialogue as incomplete.
There is, believe it or not, some evidence that it may have been intentionally incomplete (a position held by Proclus and increasingly common in modern times). In Timaeus, when Critias summarized the background for the story, we learned (21d) that Solon also failed to complete his poem on the story; one can easily see this as an anticipation. It's certainly the case that the incompleteness makes the story even more striking (which is perhaps why when Sir Francis Bacon wrote his work, The New Atlantis, he also left it incomplete in imitation). There is countervailing evidence, however; Critias strongly suggests that there will be a follow-up dialogue, with a speech by Hermocrates, but this seems to require that it itself be finished. (And yet, on the other side again, we are curiously never told what the topic of Hermocrates's speech would be.) Probably we need to read it with double vision, taking it in both ways, just to see how it changes in that light.
You can read Critias online in English at the Perseus Project and in French at Wikisource.
(in order of appearance)
Since the dialogue occurs immediately after Timaeus, the characters are the same. Timaeus opens the dialogue, transitioning from Timaeus; then Critias, Socrates, and Hermocrates.
The Plot and The Thought
Timaeus opens the dialogue by remarking that he's glad to be done with his part. Critias remarks that it is very difficult to represent human life accurately, and asks everyone for sympathy and good will, which Socrates readily grants. Hermocrates also encourages Critias to go on, and he does.
Nine thousand years ago there was a great war. Athens, which ruled the Mediterranean, fought for the entire war, and they fought against the island of Atlantis. Critias begins with how these two powers arose.
Long before, the gods freely divided the earth, raising human beings "as their own chattel and livestock, as do shepherds their sheep" (109b), but guiding us by persuasion rather than coercion. Because Hephaestus and Athena had a common nature, as brother and sister and as having common interests, they shared the same territory. They drew forth good men from the land and taught them how to build their society:
Now, at that time, the other classes of citizens who dwelt in our city were engaged in manufacture and producing food from the earth, but the warrior class that had originally been separated from them by god-like men lived apart. They had all that was appropriate to their training and all their possessions as the common property of all, and they asked to receive nothing from the other citizens beyond what they needed to live. Their activities were all the activities that were spoken of yesterday, when the guardians proposed by our theory were discussed. (110c-d)
The Athens of the day was far from the sea, since the land had not eroded through the millenia since, and it was a lush and fertile land. Athens itself was larger -- its acropolis alone was nearly as large as the Athens of Critias day.
Critias then turns to the island of Atlantis, which fell under the allotment of Poseidon. People sprang from the earth there, too, and among them was one named Evenor, with his wife Leucippe, who had a daughter called Clito. Poseidon slept with Clito, and to keep her protected, he broke the island into perfect rings. This made her hill, at the center of the island, inaccessible, for there were no ships or boats yet. The center portion of the island he made fit for a god, and begot five pairs of twin sons. He then divided the island into ten parts and made them kings, giving the eldest the central portion and the high kingship. Critias even gives the names, although he does note that they are Greek versions of Egyptian versions of whatever the originals were: Atlas, Eumelos/Gadirus, Ampheres, Euaemon, Mneseas, Autochthon, Elasippus, Mestor Azaes, Diaprepes.
The island of Atlantis was rich in all good things, so the people prospered extraordinarily. This was unsurprising, because they were partly divine. Having a natural affinity to the gods, they had mildness (praotes) and prudence (phronesis); they were sober in judgment, not intoxicated by their luxuries, although their wealth increased along with their wealth and virtue.
But as their divine portion grew faint and they became more mortal over time, they grew more disordered:
To whoever had eyes to see, they appeared hideous, since they were losing the finest of what were once their most treasured possessions. But to those who were blind to the true way of life oriented to happiness [eudaimonia], it was at this time that they gave the semblance of being supremely beauteous and blessed. Yet inwardly they were filled with unjust lust [pleonexia] for possessions and power. (121b)
Zeus decides that to punish them and "to make them more careful and harmonious as a result of their chastisement" (121c) -- although this seems a little incongruous given that we already know that the punishment will consist of destroying their entire civilization, sinking their island, and, in the process, devastating the rest of the world -- and therefore calls the gods to council. This gives us the ending of the dialogue:
And when he had gathered them together, he said
* In a notable and obviously deliberate flip of the Peloponnesian War, in which Athens was an imperialistic sea power and Sparta was a conservative and martial land power, in Critias's tale Athens is a conservative and martial land power, while Atlantis is an imperialistic sea power.
* The gods in the dialogue have a demiurgic role, creating by being Minds that persuade the world. And notice that Critias uses Timaeus's description of human nature to describe the Atlantids, often including the same terms.
Critias also uses the term pleonexia, the craving for more, to describe the degradation of the Atlantids; pleonexia is identified in the Republic, as the cause of injustice, and, indeed, almost all the features of Socrates' ideal city are designed to counter different aspects of it.
There is a common position in scholarship on the dialogue that Critias's tale falls short in some key way, that he is not, in fact, conveying Socrates' ideal city in the historic past; but both of these points seem to me to tell against such a position. Critias is, in fact, doing what he said he would set out to do: he draws on both Timaeus and Socrates to populate his story with human beings. Of course, the problem with an incomplete dialogue is that everything can be taken in any number of different ways, depending on how you assume it would have continued.
* Plutarch (who takes the Platonic attribution of the story to Solon seriously) has a nice passage in his Life of Solon (Chapter 32) on this dialogue:
Plato, ambitious to elaborate and adorn the subject of the lost Atlantis, as if it were the soil of a fair estate unoccupied, but appropriately his by virtue of some kinship with Solon, began the work by laying out great porches, enclosures, and courtyards, such as no story, tale, or poesy ever had before. But he was late in beginning, and ended his life before his work. Therefore the greater our delight in what he actually wrote, the greater is our distress in view of what he left undone. For as the Olympieium in the city of Athens, so the tale of the lost Atlantis in the wisdom of Plato is the only one among many beautiful works to remain unfinished.
Quotations are from Diskin Clay's translation in Plato, Complete Works, Cooper & Hutchinson, eds., pp. 1292-1306.