(Note: a large part of this answer is taken from an answer I have given earlier. I have added to and slightly reworded some parts of it.)
Atlantis was not real. It is a fictitious place created by Plato as an allegory.
Location
It is often referred to as an island, and it did have an island at its core. According to Plato the island was quite large:
an island greater in extent than Libya and Asia
For ancient Greeks, 'Libya' meant the coast of North-Africa and 'Asia' meant the area what is now Turkey. So that's quite a big island, a continent almost. But in Plato's stories Atlantis was not just an island - it was an empire spanning the western Mediterranean:
All these and their descendants for many generations were the inhabitants and rulers of divers islands in the open sea; and also, as has been already said, they held sway in our direction over the country within the Pillars as far as Egypt and Tyrrhenia.
(From Plato's Critias, using this translation.)
The "Pillars" are the 'Pillars of Hercules', rock formations which flank the Strait of Gibraltar - Gibraltar itself being one of the 'pillars'. The island of Atlantis itself lay beyond the Pillars, so somewhere in the Atlantic Ocean (that's where the ocean gets it's name from). "[W]ithin the Pillars" then means the Mediterranean. If the empire stretches from Gibraltar as far as Egypt and Tyrrhenia (Italy), it covered quite a large area, covering more ground than for example the Carthaginian empire. We have no evidence for such a large empire having existed there at that time. Furthermore, we have no mention of Atlantis by ancient historians such as Herodotus.
Instead of something historical, Plato's Atlantis is a description of an ideal state, something to strive for. It is comparable to Thomas More's Utopia - nobody is suggesting Utopia was a real island some time long ago.
Inhabitants
The original inhabitants of Atlantis were, of course, the gods themselves. The gods had divided the Earth amongst themselves and Atlantis fell to the siblings Hephaestus (god of artisans) and Athena (goddess of wisdom). Together they populated the island with virtuous people. Over time though, the people lost their virtue, as Plato describes in Critias (emphasis mine):
For many generations, as long as the divine nature lasted in them, they were obedient to the laws, and well-affectioned towards the god, whose seed they were; for they possessed true and in every way great spirits, uniting gentleness with wisdom in the various chances of life, and in their intercourse with one another. They despised everything but virtue, caring little for their present state of life, and thinking lightly of the possession of gold and other property, which seemed only a burden to them; neither were they intoxicated by luxury; nor did wealth deprive them of their self-control; but they were sober, and saw clearly that all these goods are increased by virtue and friendship with one another, whereas by too great regard and respect for them, they are lost and friendship with them.
If there is too much focus on worldly matters, then virtue and friendship are lost. Of course, this is exactly what happened to the Atlanteans (emphasis mine):
By such reflections and by the continuance in them of a divine nature, the qualities which we have described grew and increased among them; but when the divine portion began to fade away, and became diluted too often and too much with the mortal admixture, and the human nature got the upper hand, they then, being unable to bear their fortune, behaved unseemly, and to him who had an eye to see grew visibly debased, for they were losing the fairest of their precious gifts; but to those who had no eye to see the true happiness, they appeared glorious and blessed at the very time when they were full of avarice and unrighteous power.
Sinking of Atlantis
Atlantis was not used by Plato only to describe an ideal state. In another of his works, Timaeus, we find the famous sinking of Atlantis. But Atlantis did not sink until after the Athenians had fought off the Atlanteans. As said above, mythical Atlantis ruled the western Mediterranean and was expanding east. In this way, Atlantis came into direct conflict with Athens:
and then, Solon, your country [Athens] shone forth, in the excellence of her virtue and strength, among all mankind. She was pre-eminent in courage and military skill, and was the leader of the Hellenes. And when the rest fell off from her, being compelled to stand alone, after having undergone the very extremity of danger, she defeated and triumphed over the invaders, and preserved from slavery those who were not yet subjugated, and generously liberated all the rest of us who dwell within the pillars.
Athens defeated Atlantis and apparently single-handedly destroyed its empire. Again, we see virtue playing an important role here. It was the 'excellence of her virtue and strength' which brought victory. After Athens' victory, violent earthquakes and floods struck Atlantis, and the island sank into the sea.
Conclusion
So both times Plato mentions Atlantis it is to emphasise virtue. There is no suggestion at all that all this has really happened. It is an allegory, a tale to educate people on what moral qualities are important.