The Atlantis story is a marvelously mutable myth, susceptible to being plucked out of context and pasted into any convenient theory about the unknown and unknowable.
Classicists agree that Atlantis is a parable - a cautionary tale devised by Plato as an illustration of how not to govern. It was understood as such by his first generation of students, including Aristotle, who once observed that Atlantis was destroyed by the author who invented it (Strabo 13.598). The idea that Atlantis was real emerged a generation later, when Crantor - the first commentator on the works of Plato - claimed that Atlantis had been an actual place. There are only stray mentions of Atlantis, mostly casual, in later classical literature. But by the fifth century CE - when the ascendancy of Neoplatonism made Plato's late dialogues a topic of frequent discussion and debate - there were apparently several schools of thought about Atlantis among Platonist philosophers, with some believing that it had been real. All this, however, was purely academic. There is no ancient evidence for widespread popular belief in Atlantis.
The idea that Atlantis was real reemerged after the European discovery of the New World, when various humanists identified the Americas with Plato's lost civilization (or, since the Americas were inconveniently not underwater, as lands colonized by Atlantean survivors). There were also a few early modern attempts to claim that Atlantis was actually in/beneath/near various European countries. But the modern conception of, and widespread belief in, Atlantis has its roots in the late nineteenth century.
It all began with Jules Verne's wildly popular 20,000 Leagues under the Sea (published in 1870), which features a scene in which Aronnax and Nemo take a quick scuba tour of the ruins of Atlantis:
"In fact, there beneath my eyes was a town in ruins, demolished, overwhelmed, laid low, its roofs caved in, its temples pulled down, its arches dislocated, its columns stretching over the earth; in these ruins you could still detect the solid proportions of a sort of Tuscan architecture; farther off, the remains of a gigantic aqueduct; here, the caked heights of an acropolis along with the fluid forms of a Parthenon; there, the remnants of a wharf, as if some bygone port had long ago harbored merchant vessels and triple-tiered war galleys on the shores of some lost ocean; still farther off, long rows of collapsing walls, deserted thoroughfares, a whole Pompeii buried under the waters, which Captain Nemo had resurrected before my eyes! Where was I? Where was I? I had to find out at all cost, I wanted to speak, I wanted to rip off the copper sphere imprisoning my head. But Captain Nemo came over and stopped me with a gesture. Then, picking up a piece of chalky stone, he advanced to a black basaltic rock and scrawled this one word: Atlantis."
Atlantis had never been forgotten; but prominent mention in an international bestseller got people talking - and writing - about Plato's story again. The real turning point came twelve years after 20,000 Leagues was published, when an American politician-cum-pseudo-scientist named Ignatius Donnelly published Atlantis: the Antediluvian World. As Donnelly says in the first chapter:
"This book is an attempt to demonstrate several distinct and novel propositions. These are:
That there once existed in the Atlantic Ocean, opposite the mouth of the Mediterranean Sea, a large island, which was the remnant of an Atlantic continent, and known to the ancient world as Atlantis.
That the description of this island given by Plato is not, as has been long supposed, fable, but veritable history.
That Atlantis was the region where man first rose from a state of barbarism to civilization.
That it became, in the course of ages, a populous and mighty nation, from whose overflowings the shores of the Gulf of Mexico, the Mississippi River, the Amazon, the Pacific coast of South America, the Mediterranean, the west coast of Europe and Africa, the Baltic, the Black Sea, and the Caspian were populated by civilized nations.
That it was the true Antediluvian world; the Garden of Eden; the Gardens of the Hesperides; the Elysian Fields; the Gardens of Alcinous; the Mesomphalos; the Olympos; the Asgard of the traditions of the ancient nations; representing a universal memory of a great land, where early mankind dwelt for ages in peace and happiness.
That the gods and goddesses of the ancient Greeks, the Phœnicians, the Hindoos, and the Scandinavians were simply the kings, queens, and heroes of Atlantis; and the acts attributed to them in mythology are a confused recollection of real historical events.
[more of the same]
That Atlantis perished in a terrible convulsion of nature, in which the whole island sunk into the ocean, with nearly all its inhabitants.
That a few persons escaped in ships and on rafts, and carried to the nations east and west the tidings of the appalling catastrophe, which has survived to our own time in the Flood and Deluge legends of the different nations of the old and new worlds."
Employing a weighty - and to many of his contemporaries, compelling - blend of misapplied scientific evidence and decontextualized archaeology, Donnelly proceeded to argue that Atlantis (a large island off the coast of Spain) had (a) originated just about every advance in early human civilization, (b) colonized the choicest bits of the Americas and Eurasia, and (c) vanished, leaving only memories (disguised, often as not, as myths) of its grandeur.
Donnelly's book was a runaway bestseller (Darwin was skeptical, but Gladstone loved it), and permanently imprinted Atlantis on the popular imagination. Although his scientific and historical claims were soon discredited, the idea that Atlantis could be real proved enduring. Atlantis was picked up by Madame Blavatsky and the Theosophists, who wove the lost civilization into their labyrinthine theories of root races. The American psychic Edgar Cayce (AKA the sleeping prophet) drew from both Donnelly and Blavatsky in his weird imaginings of Atlantis, which featured super-powered crystals and a surprising amount of bestiality. Even the Nazis took a swing at Atlantis - Himmler's Ahnenerbe sponsored an expedition to the Canaries to search for the remains of an (Aryan) Atlantean Civilization.
For a brief discussion of all this, feel free to check out my two videos on Atlantis, both watchable here. The first discusses Plato's descriptions of Atlantis; the second provides additional detail on some of the ground covered in this post.