Sea. Saagar means sea, and Manthan means churning; so Saagar Manthan means churning of the sea. This event is especially referred to Saagar Manthan done by Devtaa and Daitya. Saagar's synonyms are Jaladhi,  Paathodhi,  Ratnaakar,  Samudra,  Sindhu,  Udadhi.

Once a war broke between Devtaa and Daitya. Daitya defeated Devtaa and took control over Swarg Lok. Indra had lost his wealth also because of Durvaasaa Muni's Shaap. So he went to Vishnu for His help. Vishnu said - "This time is not good for you. Keep patience and extend friendship with Daitya. Instigate them to churn the sea (Kshee Saagar). Both Lakshmee and Amrit will come out of it. I will help you to get both of them. Use Mandaraachal Parvat as the churning rod and Vaasuki Naag as the churning rope." Indra went back and went to Daitya Raaj Bali to extend his friendship and propose the churning of the sea. Bali agreed and both Devtaa and Daitya started churning the sea.

Devtaa were at the tail side of Vaasuki and Daitya were at the mouth side of Vaasuki. In the absence of any base, Mandar was shaking violently and sinking in the sea. So Vishnu assumed the form of a Koorm (tortoise) and lifted Mandaraachal on His back. During the churning 14 Ratn (Jewels) came out of the sea -

(1) Halaahal Vish (poison) - Shiv drank it and kept it in his throat
(2) Surabhi Cow - Rishi took it
(3) Uchhshraivaa Horse - Raajaa Bali took it
(4) Airaavat Elephant - Indra took it
(5) Kaustubh Mani - Vishnu wore it on His heart
(6) Paarijaat Tree - Indra took it and planted it in his Nandan Van
(7) Apsaraa - Indra took them
(8) Lakshmee - She chose Vishnu as her husband
(9) Vaarunee Devee - Daitya took her
(10) Shaarng Bow - Vishnu took it
(11) Chandramaa - Shiv took it and kept it on his head
(12) Shankh - Paanchjanya conch shell, Vishnu took it
(13) Dhanvantari - Traditional doctor
(14) Amrit - Ambrosia

There is a Shlok about the 14 Ratn coming out of the sea, I don't know how far this Shlok is correct, but I can say it for sure that Garud (Baaz) did not come out of the sea, he was the son of Kashyap and Vinataa [I got this Shlok from Manas Pant] --
Shree Rambhaa Vishu Vaarunee Ameeya Shankha Gajraaj
Dhanwantaaree Dhanu Dhenu Taru Chandramaa Mani Baaz.

As the Amrit came out both Devtaa and Daitya ran to take it. Daitya could take the lead and ran away with it. Now, Vishnu assumed the form of a beautiful woman, Mohinee, and appeared among Daitya. All Daitya got attracted to see her and asked her to be their wife and distribute Amrit to them. Mohinee agreed and took the pitcher of Amrit, but She did not give Amrit to Daitya, rather She gave it to Devtaa.

A Daitya named Raahu was noticing all this, he assumed the form of Chandramaa and got mixed with Devtaa. Mohinee gave some Amrit to him too, but at the same time Soorya and Chandramaa recognized him and complained about him to Mohinee. Vishnu took out His Chakra immediately and cut his neck. But by that time he had already gulped down some Amrit, so he did not die, rather got divided in two - his head was Raahu and his trunk became Ketu (tail). Since then he has developed an enmity with Soorya and Chandramaa, he swallows them on Amaavasyaa (No Moon day) and Poornimaa (Full Moon day) respectively. That is why we have solar and lunar eclipses.

Padm Puraan, 2/15 (p 324), says that four women came out from the sea at the time of Saagar Manthan - Lakshmee, Vaarunee, Kaamodaa, and Jyeshthaa. Kaamodaa appeared from the wave of Amrit. She will assume the form of a tree to please Vishnu Bhagavaan and will always give pleasure to Vishnu. In this form she will be known as Tulasee. Jagannaath will always enjoy with her. Whoever will offer Him even one leaf of Tulasee, He will think about giving him "What can I give him?" and He will be very pleased with him.

Padm Puraan, 5/37 says that Saagar Manthan started on a pure Ekaadashee Tithi.

[Mahaabhaarat, 0/6] After a while a mild Moon came out; then came out Lakshmee dressed in white; then Som; then the white horse; then Kaustubh Mani which adorned Naaraayan's chest. Then Lakshmee, Som and the horse came to Naaraayan. Then came out Dhanvantari himself with white vessel of nectar. Seeing it Asur cried, "It is ours, it is ours." After it came out the huge elephant Airaavat with two pairs of tusks. Indra took him. Then appeared Kaalkoot poison. At the request of Brahmaa Jee, Shiv swallowed that poison. He kept it in his throat. It is said that he was called Neelkanth from the same time.

Bhavishya Puraan, 1/21 says that five cows came out of the Saagar Manthan - Nandaa, Subhadraa, Surabhi, Sumanaa and Shobhanaavatee.

Where Its Reference is Found
The Legend of the churning of the ocean and of its entities 'jewels' emerging there from appears to be very popular in post-Vaidik Sanskrit Literature, The legend is not found in Vaidik Literature, it is found in the two Epics - the Mahaabhaarat and the Raamaayan and in the Puraan literature. below is a comparative study of the legend as it occurs in the following works :

(1) The Mahaabhaarat (Mbh). [The critical edition published by the Bhnmlarkar Oriental Research Institute, Poonaa].
(2) The Raamaayan (R) [The Vaalmeeki Raamaayan critical edition, published by the Oriental Institute, Baroda].
(3) The Matsya Puraan (M) [Shri Venkateshwara Press, Bombay],
(4) The Vishnu Puraan (Vi) [Edited and published by TR Vyasacharya, printed at the Nirnaya Sagar Press, Bombay, 1914].
(5) The Brahmaand Puraan (Br.) [Shri Venkateshwara Press, Bombay],
(6) The Padm Puraan (P; [edited by V. N. Mandalik, Anandashrama, Poona].
(7) The Bhaagavat (Bhag) [published by the Nirnaya Sagara Press, Bombay; Gita Press, Gorakhpur].
(8) The Aagneya Mahaauraan (A) [Shri Venkateshwaia Press, Bombay].
(9) The Skand Puraan (SK). [Shri Venkateshwara Press, Bombay],

It has also been discovered that the legend, as it is told from work to work, differs in many respects in its several parts. For instance, there
is no unanimity with regard to the number of entities which emerged from the ocean, or with regard to the sequence in which they emerged. (See some of lists here) In order to highlight the differences in the legend as it develops from one part to another, the story is
divided for the purpose of present study into the following sits sections :
(1) The occasion for the story of (the churning of the ocean for Amrit) Amrit Manthana (AM).
(2) The motive of the AM.
(3) Preparations for the AM.
(4) The churning of the Ocean for AM.
(5) Emergence of Amrit and of other entities.
(6) Vishnu's intervention in the fight for Amrit, to the advantage of the gods. This plan of division would, it is hoped, facilitate the study of the points of similarity and difference, as the legend develops from one part to another. Besides, with a view to bringing out the significant points, if any, during this comparative study, there are inserted in between within rectangular brackets notes by the present writer.

It will be noticed that in the following sections giving the development of the legend from one work to another, the Padm Puraan (P) has been referred to twice. It is because the Padm Puraan gives two different accounts of the legend in two different places viz, in its Srishti Khand and Uttar Khand respectively. These two passages from Padm Puraan are therefore given a separate place in the following sections. P (Srishti Khand) and P (Uttar Khand) are abbreviated to P (S) and P (U) respectively and are referred to as such in the sequel.

(1) The Occasions for the Story of AM
[Mk 1.15. 1-4]
While narrating the story of the two sisters Kadroo and Vinataa as a part of the Aasteek legend, Soot Jee said to Shaunak : "The two sisters beheld from near the divine horse Uchchaishravaa the best of the horses which arose while the Amrit was being churned out." Thereupon, Saunak Jee asked Soot Jee to tell him how and where the Amrit was churned out during which process was born the illustrious horse. Soot Jee then takes this occasion to tell the story of AM.

[Raamaayan. 1-44.8-13 ; 1-45. 1-6 ; 1-46.10]
Raam seeing the city of Vishaal on the bank of the Gangaa, asked Vishwaamitra to tell him the previous history of that city. Vishwaamitra told Raam that the site where the city of Vishaal stood had been formerly the scene of great austerities practiced by Diti, the wife of Mareechi. As a part of the story as to why Diti practiced austerities, Vishwaamitra tells Raam the episode of AM in which the sons of Aditi got the Amrit and killed the sons of Diti who grieving over the loss of her sons, practiced penance in order to get a son who would conquer Indra and the three worlds.

Matsya Puraan. 249. 1-3
The sages, hearing from Soot Jee 'about the greatness of Naaraayan, asked him to tell them how gods attained immortality. Soot Jee begins with saying that it was the two Gods Naaraayana ami Matadeva who had helped the gods to attain immortality and proceeds towards the story of the AM.

Vishnu Puraan, 1. 9. 1-75
Paraashar introduces the story of AM in answer to Maitreya Muni's quests as to how V W came by fa or Lakshmee. Once Durvaasaa Rishi happened to see Indra on his way. he had a flower garland in his hand, so considering Indra a worthy man to possess it, he threw that garland at Indra. Indra placed the garland on the head of his elephant Airaavat. The Airaavat threw down the garland. Durvaasaa, enraged at this insult, cursed Indra to the effect that the latter would be deprived of his kingdom of the three worlds. Indra and the gods, thus deprived of their glory went along with Brahmaa to Vishnu to pray to him for restoring them to their former splendor".