After the Pralaya, or deluge, the Devas were stuck with misfortune. Sage Durvasa had presented Indra with Santanaka, the garland that never fades. Indra, however, ignored it and incurred the wrath of the asetic. Durvasa cursed the Devas to lose their strength. This severely hampered the Devas in their never ending war against the asuras. Of course, the Devas knew the trick of Sanjivani - the art of bringing the dead back to life - but since they had learnt this trick from the asuras in the first place (using trickery, of course!), it was not any great advantage. Something else was required. Brihaspati, the guru of the Devas, went to Vishnu requesting his help. Vishnu suggested that the Devas churn the ocean of milk for Amrita, or the nectar of immortality. But the problem here was that the Devas could not do it alone. The Devas could do it if the asuras helped them, but that raised the potential problem of the asuras not giving the Devas any of the Amrita once it had been churned out. However, since Vishnu assured the Devas that he prevent that from happening, the Devas duly enlisted the help of the asuras. The asuras were skeptical, that being their very nature, but overcome by greed, they agreed to help, confident that if push came to shove, they could easily defeat the Devas in a battle.
The mightly mountain Mandara was enlisted as the churning rod, while Vasuki, the king of serpants, agreed to become the rope. Vishnu, being the only God who could match the asuras cunning, asked the asuras to hold the hood of Vasuki while churning, so that it would befit their stature. The asuras gleefully agreed. But this was sheer trickery, since Vasuki frequently let out fumes that weakened the asuras. The Devas were left to hold the tail end of Vasuki, and the churning duly began. But since Mandara did not have any support, it started sinking into the ocean. Realizing the problem, Vishnu quickly incarnated as a tortoise - the Coorma avatar - and supported the Mandara mountain on His back. With the support provided, the churning was resumed.
The first thing to emerge from the churning was Halahala, or Kalakuta - the worst poison in the universe. This immediately created a panic among the Devas and asuras, and they began to choke on the poison. It required Shiva to intervene and swallow the poison. However, Parvati held his throat and stopped the poison from going into his body. This made Shiva blue from the throat up, and hence he has also been referred to as Neelakanta from then on.
As the churning continued, more celestial objects emerged from the ocean. Suri (the Goddess of wine and intoxication, who spread all over the world), Surabi (the celestial cow, which the sages took for their religious rites), Parijatha (the divine wish fulfilling tree that went to heaven), the Apsaras (celestial nymphs), Chandra (the Moon), Kaustubha (the precious gem), Ucchaishravas (the divine horse), Airavata (the four-tusked elephant), Panchajaya (the conch), Sharanga (the invincible bow).
After this, another set of poisons emerged, but none were of the same intensity as Halahala. The snakes took them for their own. And then suddenly, Lakshmi, the Goddess of Fortune, emerged on a lotus blossom, wearing a lotus garland and holding a lotus in her hand, and smiling radiantly. She immediately selected Vishnu and resided on His chest.
Finally, Dhanvantari, the physician of the Devas, emerged, holding the pot of Amrita in her hand. The asuras quickly snatched the nectar from her hand before the Devas could react and were rejoicing over their prize, when the Devas appealed to Vishnu to intervene. Vishnu assumed his dual form of Mohini, the female form of such immense beauty that could cause even Shiva to lose control of his senses. She offered to distribute the amrita evenly between the Devas and asuras.
Mohini, being none other than Vishnu, had the same cunning as Him, and served the Amrita to the Devas first, and served intoxicants to the asuras by a sleight of hand. But the asuras soon realized what had happened and quickly geared up for a fight. The Devas, however, were already immortal and easily vanquished the asuras. One of the asuras, Rahu, was smarter than the rest, and assumed the form of a serpent and began licking the last few drops of the Amrita. The Sun and Moon Gods saw this and informed Vishnu, who quickly cut off the head of Rahu before the nectar could pass through his throat. The head, however, remained immortal and Rahu periodically swallows the Sun and Moon in revenge, but since his throat is open-ended, they come out soon from the other end.
Well, so thats that for the story of the Coorma avatar. This is the story, as narrated with an eye only on the superficial details. This doesn't even begin to describe the underlined deeper meaning. Almost all Hindu texts are written in this manner - on the surface is a story that makes for nice reading, is often taken only literally, and occasionally as true. But every one of these stories has a deeper meaning that goes beyond the superficial nature of the story. Every tiny detail has been written not with an eye for aesthetic sense, but with deep wisdom. A study of this wisdom in the texts would probably take many lifetimes (given that the body of Hindu scriptures is really vast).
The symbolism of the Coorma avatar is the best example of symbolism that I have found in my limited reading (and probably the most significant), and this is described in the next post. I'll explain the symbolism as best as I know, and apologise for errors in advance.