This day is celebrated as Kurma Jayanti. The Kurma Puran authored by Ved Vyas, details narrations of the various pastimes of this incarnation of the Lord.
Various other scriptures such as the Ramayan and Mahabharat too detail the episode of Samudra Manthan - Churning of the ocean.
In the eighteenth chapter of Vasudev Mahatmya the Lord says, 'When the sons of Kashyapa churn the ocean with the Mandara Mountain as the churning rod, which has no base to rest on, I will carry that Mandara on my back, assuming the form of a great tortoise'. (Verse 19)
In this incarnation, the Lord Vishnu manifested as a tortoise. Due to Sage Durvasa's curse, the demi-gods lost all their wealth. The sage out of respect presented a garland of flowers to Indra who carelessly ignored his humbleness. Indra's elephant trampeled the garland, hence the sage in rage cursed Indra to loose all his wealth in to ocean. You can read more in detail about this legend within Shree Vasudev Mahatmya
The Devtas approached Vishnu for help. Vishnu then asked them to churn the ocean of milk after adding medicines into the ocean. Mt Mandara could be used a the churning stick he said. He requested them to ask them help of Asuras in lifting the mountain in exchange for offer of the share of nectar of immortality that would ensue from the churning. Both the devatas and the asuras churned the ocean using the serpent Vasuki as the rope. At the start, playing a Machiavellian trick, Indra, king of the gods asked the asuras for the head end of vasuki. But asuras suspecting foul play, took the head end, only to be deceived as the poison from Vasuki was slowly weakening them. But as churning was proceeding the mountain was sinking and then Lord Vishnu took the form of the tortoise KURMA and kept the mountain afloat. As soon as the bowl of amrita, the nectar of immortality was out, the asuras grabbed it. Then Lord Vishnu took the form of an apsara, a beautiful maiden, and seduced the asuras into letting her distribute the nectar and also to abide by her order of distribution. As soon as the devatas were served the maiden disappeared thus totally deceiving the asuras and making them totally weak.
In the 3rd Vachanamrut of Gadhda Pratham Prakan the Lord says, ‘“Even a person who can constantly see the form of the Lord in his heart, should recall the pastimes performed by Him in His different incarnations in different places.