Matsya Avataar is Vishnu's first incarnation. Matsya Avataar has been variedly narrated in different texts. The Shatpath Braahman and the Bhaagavat Puraan relate it to Manu, the Padm Puraan relates it to demon Shankhaasur, and the Matsya Puraan relates it to demon Hayagreev.

Bhaagvat Puraan Version
According to Bhaagvat Puraan, 8/8, Hayagreev name Asur took away Ved from Brahmaa's mouth when he was going to sleep after the Pralaya. So to establish Dharm again Vishnu took fish form and appeared in a river where King Satyavrat was taking bath. As he was about to water Arghya to Soorya, that fish came into his hands. He tried to put it in water again but the Fish requested him not to do so. He brought it home and put it in a pitcher. The fish continued to expand over night, till the King had to throw him in the sea. The King asked - "You don't look like a common fish, who are you?" The Fish said - "O King, You are right. I am Vishnu. On the 7th day from today the Pralaya is going to happen. This Prithvi will submerge in the water. Till then you collect the subtle bodies of all the species, Saptarshi, and all kinds of seeds.

On the 7th day you will see a boat coming to you. You sit in it, it will float on the water throughout the Pralay water. Don't be afraid, the Sapt Rishi will be with you. After a while I will appear in the same form as you are seeing me now. I will have one horn. You take Vaasuki Naag and tie him with my horn. I will take you through out the period of Brahma's sleep. There will be no light you will see only by the Divine light of Rishi. At that time whatever you will ask I will answer to you and deliver sermons to you too." After saying this Bhagavaan disappeared. The same thing happened as Bhagavaan told the King Satyavrat. All this is narrated in in Matsya Puraan.

After Brahmaa woke up, Bhagavaan killed Hayagreev and brought back Ved from him.

Padm Puraan Version
This story comes from Padm Puraan, 5/12 - "Satyabhaamaa asked Krishn - "Why do you love Kaarttik Maas and Ekaadashee Tithi, tell me the reason of this?" Krishn said - "Listen, In earlier times, King Prithu asked Naarad Jee a similar question. Naarad Jee said - "In earlier times there was an Asur named Shankh. He was the son of Samudra. He defeated all Devtaa, so Devtaa hid in caves. He thought that these Devtaa are powerful because of Ved, so let me take away their Ved, this will make them weak." At that time Chaaturmaasya (Shraavan, Bhaadrapad, Aashwin and Kaartik) was going on and Vishnu was sleeping in His Ksheer Saagar. Before He went to sleep all Adhi-Daivik forms of Ved also merged into the Divine water of Ksheer Saagar, that is why Devtaa were also very weak.

That is why Brahmaa Jee went to Vishnu along with other Devtaa and prayed Him. Pleased with their worship, Vishnu said - "On Kaartik Shukla Prabodhinee Ekaadashee day, when only one Prahar (3 hours) night remains, whoever will worship me, they will come near me. Shankhaasur has kept all Ved in water. I will kill him and bring the Ved from him. From today, every year Ved and I will rest in water in Kaartik Maas. Since you have come on Ekaadashee, that is why this Tithi will be very dear to me. If anyone takes up Ekaadashee and Kaarttik Maas Vrat, he will surely come to me."

Thinking that Devtaa were weak, Shankhaasur brought the Ved from there and hid in the sea. The absence of the Ved, which had been guiding mankind to the path of righteousness, caused an outbreak of all kinds of sins and vices and pervaded all aspects of human life. Shankhaasur was hiding with the Vedas into deep waters inside a conch named Panchajana. After killing Shankhaasur, Vishnu brought his body Shankh (Conch) along with Him and since then He has been carrying it as the Paanchjanya conch as one of his attributes. When He incarnated as Krishn in Dwaapar Yug, He blew it to announce the commencement of the Mahaabhaarat war.