MASTYA Avatar
In MASTYA Avatar, Lord Vishnu incarnates himself as a fish in this world. In the earliest yuga (era) of Sata-yuga, a king named Manu was performing severe penance for thousands of years. One day as he was performing ablutions with river water, a small fish came into his hands and just as he was about to throw the fish back into the river, the fish requested the king to save its life. Heeding its request, the king put the fish into a jar of water but the fish started growing and the jar was not big enough for it. Then the king threw it into the river, but it soon it outgrew the river and the king then threw it into Ganges and then into the ocean. The king realised that it was Lord Vishnu himself and then the lord made an appearance and made a special request to the king. It predicted that the world would come to an end by a huge flood in seven days and requested the king to build a huge boat and take the seven sages(hermits), seeds of all plants, one animal of each type and told him that he would appear as a fish to propel the boat to Mt Himavan for surviving the flood to the next yuga(eon). True to his word, after seven days the Lord appeared and the king tied the boat to the fish by using the royal serpent Vasuki and the fish took all of them to Mt Himavan and kept them there till the flood was over and in the new era, the king started procreation a for the new era.
KURMA Avatar
In KURMA Avatar, Lord Vishnu incarnates himself as a tortoise. It is an interesting story involving both the gods (Devtas) and asuras (demons). In the ongoing saga of battle between the gods and asuras, on one occasion the gods suddenly lost all their strength due to a curse by the short-tempered sage Durvasa. The sage had once presented a garland of flowers to Indra,king of gods, who carelessly gave it away to his elephant which trampled it.
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.
VARAHA Avatar
In VARAHA Avatar, Lord Vishnu incarnates himself as a boar in this world. A demon Hiranyaksha, had prayed for Lord Brahma and got awarded a boon that no beast nor man nor god could kill him. But somehow from the list of beasts the name of boar was missing. This proved to be his lacunae. He then started a campaign of plunder across the worlds. He pushed the world to the Pataal loka, or the under of the sea. He stole the Vedas, the holy scriptures from the Lord Brahma, while he was asleep and performed huge atrocities.
To retrieve the Vedas and to save the world the Lord Vishnu assumed the role of a boar and brought out the earth from the under of the ocean, using its two tusks. It then killed Hiranyaksha and retrieved the Vedas from the asura and brought it back to the safe custody of the Lord Brahma.
There are some alternative versions of the story of Varaha too.
NARASIMHA Avatar
In NARASIMHA Avatar, Lord Vishnu incarnates himself as a semi-man,semi-lion in this world. The king of demons(asuras), Hiranyakasyapa, wanted to become immortal and wanted to remain young forever. To this end, he meditated for Lord Brahma and because of his severe penance, the gods were frightened and asked Brahma to pacify the king. Brahma was impressed by his austerity and granted him a wish. HiranyaKasyapa wished that he be neither killed by a man or beast, nor in daylight or at night and neither inside or outside a building. Having obtained the wish he considered himself the supreme God and frobade all worship of gods by anyone. But his son Prahlada, was an ardent devotee of Vishnu. This enraged Hiranyakasyapa very much. He ordered numerous ways to kill Prahlada including asking his sister Holika to sit with Prahlada in the fire. But everytime Prahlada escaped unhurt. Enraged, once he asked Prahlad to show him the Lord Vishnu. Prahlad said, "He is everywhere". Further enraged, Hiranyakasyapa knocked down a pillar, and asked if Lord was present there. Lord Vishnu then emerged as a half lion, half man from the pillar which was neither inside the house nor outside, and the time was evening, neither night nor day. He then killed Hiranyakasyapa thus saving the life of his devotee Prahlada.
So goes the story of Hiranyakasyapa and the NARASIMHA avatar.
VAMANA Avatar
In VAMANA Avatar, Lord Vishnu incarnates himself as a dwarf priest in this world. BALI, the grandson of Prahlada was a very valorous and mighty asura. By his penance and might, he conquered the whole world. Indra and other gods fearing that he and asuras would conquer all the three worlds, went to Lord Vishnu for help. Lord Vishnu was then born as a dwarf Vamana in the household of a brahmana(priest). He went to Bali on growing up and asked for alms. Bali was delighted to offer him anything he requested even though his priest warned him that it was Lord Vishnu.
Vamana then requested for the amount of land that could come under his three feet. Bali gracefully agreed. Lord Vishnu then grew in size and covered the earth and heaven in two stride. And due to lack of space, he put his third leg on Bali himself and crushed Bali to the nether or the Patala loka(underground world), thus helping the Gods out.
PARASURAM Avatar
Parsu is the name of an axe-like weapon adopted by this incarnation.
Parasuram was a brahmin who manifested himself at the close of the satya-yug. This appearance was for the purpose of defeating the tyranny of the power-drunk kshatriyas.
When the kshatriya kings of the earth and their ministers became very corrupt and tyrannical, the goddess Prithvi (Mother Earth) went to Lord Vishnu and prayed for relief. Lord Vishnu answered the prayer and appeared on the earth as a descendant of the great sage Brigu. Parasuram avenged the gruesome murder of his father by a kshatriya king and freed the earth of oppression perpetrated by the rulers by clearing the earth of kshatriyas.
RAMA Avatar
In RAMA Avatar, Lord Vishnu incarnates himself as RAMA , the central character in the epic RAMAYANA. In this biggest epic of Indian mythology a myriad of topics are covered, including war, love, brotherhood,ideal man , son etc. It is essentially the story of an ideal king, son, father and man, RAMA. As an ideal son, he agrees to abide by wishes of his father King Dasaratha to spend 14 years in forest, along with his wife Sita and brother Lakshman. During the course of his stay in forest, the demon Ravana abducts his wife Sita. Ram then sets out in search of his wife. During this he makes friends with king of monkeys Sugreeva, and his devotee Hanuman. In the end he wages a war with Ravana and rescues Sita to return to Ayodhya and rule it as an ideal king for 1000 s of years.
KRISHNA Avatar
In KRISHNA Avatar, Lord Vishnu incarnates himself as KRISHNA , the central character in the epic MAHABHARATA. In this biggest epic of Indian mythology a myriad of topics are covered, including war, love, brotherhood, politics etc. It is essentially the story of two warring groups of cousin brothers, the PANDAVAs and the KAURAVAs. As a part of the Mahabahrata, during the war KRISHNA, gives a long discourse to his disciple ARJUNA, collectively termed as Bhagvad-Gita. Krishna, during his child-hood was responsible for the killing of Kansa. Krishna is also considered to be an ultimate playboy who was resonsible for charming all gopikaas(cowherdesses) around him.
Unlike Ramayana, Mahabharata deals with more down to earth issues like politics, human nature, human weaknesses, and does not attempt to idealise the characters as in RAMAYANA.
BUDDHA Avatar
In BUDDHA Avatar, Lord Vishnu incarnates himself as BUDDHA, the ascetic prince who renounced the throne to lead the world on the path of peace. He is the founder of the BUDDHIST religion prominent across the world. In certain sects of Hinduism, he is considered to be a divine incarnation of Lord Vishnu. He was born the crown prince of the Kapilavastu to Suddhodana and Maya. He was named Siddhartha, meaning "All thing fulfilled" by the king. But his mother died soon after his birth but Prajapati, the sister of Maya, brought Siddhartha up.
Buddha was saddened by death of living creatures, since his childhood days and used to question: "Alas! Do all living creatures kill each other?". He wasn't happy with any answers that were provided to him and he decided to find out the meaning and the absolute truth and he left his wife and child to a hermit's life in the forest and one day, became the enlightened one. His preachings spawned off the religion of Buddhism now popular across the whole world.
KALKI Avatar
In KALKI Avatar, Lord Vishnu will incarnate himself as KALKI, the machine-man, who will come riding his white horse and with his blazing sword in his hands. This is supposed to be a future avatar of Lord Vishnu. At the end of Kali Yuga (present eon) He will punish all evil doers in this world, destroy this world supposedly and recreate a golden age again. KALKI is the last of the avatars of Lord Vishnu.