The stories of 24 Avatars of Bhagwan Vishnu

It is believed “Avatar” or divine descendent refers to the Supreme Divine taking a brief human form for his own purposes, which are considered to refer to the welfare of the world which is His own creation. Potentially, Jiva in every being is in the sense an Avatar, which is best expressed by the Mahavakya, Tat Tvam As.
 
Bhagwan Vishnu have already incarnated on earth 23 times, while the 24th avatar is yet to be known as 'Kalki Avatar'. Out of these 24 avatars, 10 avatars, called as Dashavatar, are considered to be the main incarnations of Vishnu. This is Matsya avatar, Kurma avatar, Varaha avatar, Narasimha avatar, Vamana avatar, Parashuram avatar, Rama avatar, Krishna avatar, Buddha avatar and Kalki avatar. Of all the Avatars, Shri Krishna alone is considered a Purna Avatar, a direct and complete representation of the Supreme Divine Himself.
 
Spiritual Significance:

1. All the four sages, the first incarnation were symbol of brahmacharya (  ब्रह्मचर्य, state of continence and chastity) and on path salvation. Brahmacharya purify conscience (i.e. manas, bhuddhi, chit, ahankar).
2. Varaha, the second incarnation is symbol of Santosh (contentment). One should remain contented in all situations.
3. Narad ji, the third incarnation is symbol of bhakti (unselfish love). After practicing Bhrahmacharya and Santosh (संतोष, contentment), bhakti sprouts.
4. Nar-Narayan, the fourth incarnation symbolizes that when bhakti flourishes, realization of almighty is done.
5. Kapil muni, the fifth incarnation was symbol of gyan (spiritual wisdom) and vairagya (detachment). Bhakti stabilizes with spiritual wisdom and vairagya.
6. Dattatreya, the sixth incarnation reflect that when five attributes viz. brahmacharya, Santosh, gyan, bhakti and vairagya, come in life, one will gunateet (गुणातीत, beyond all properties).
7. Yagya, the seventh incarnation is a ritual performed to please the gods.
8. 11th, 12th & 13th incarnations of Kurma, Dhanvantri and Mohini were for Samundra Manthan.
9. 14th incarnation of Narsingh & 17th incarnation of Sri Hari gave the message that Bhagwan protects his devotee against all odds.
10. In 16th incarnation of Hayagreeva, Bhagwan protected Vedas.
11. In 19th incarnation of Vyas ji, knowledge of Vedas was spread for well-being of man-kind.
12. In 18th incarnation of Vamana, 21st incarnation of Shri Rama and 22nd incarnation of Shri Krishna, Bhagwan re-established Dharma by winning over negative persons / Rakshasas.
13. In 23rd incarnation of Buddha, Lord Buddha preached to stop life violence.
14. Interestingly the Avatars take different forms: starting with a sequence of animal forms suggestive of an evolutionary order:

 
Matsya a Fish, Kurma a Tortoise, Varaha a Boar, through Nrsimha, a half-lion Man, making a transition to the human sequence of Vamana, Parasurama, Rama, Krishna and Kalki making the ten accepted by the Vaishnava tradition.
 
Stories of 24 Avatars are as under:

1. First incarnation was as four sages named Sanak, Sanandan, Sanatan and Sanatkumar. 
2. Second incarnation was Varaha, a boar.  As per story, in ancient times when Lord Hiranyaksha took the earth and hid it in the sea, then Lord Vishnu appeared in the form of Varaha from the nose of Brahma. Lord Varaha started searching for the earth. With the help of his snout, he found the earth and by going inside the sea and placing it on his teeth, he brought the earth out. When Hiranyaksha demon saw this, he challenged the Varaha form of Lord Vishnu to battle. A fierce battle ensued in both. Finally Lord Varaha killed Hiranyaksha. After this, Lord Varaha pillaged the water with his hooves and set the earth on it.
3. Third incarnation was Narad ji. Narad ji is one of the seven Manas sons (psyche sons, by process of thought) of Brahma. He has attained the rank of Devarshi by hard penance. He is considered one of the exclusive devotees of Lord Vishnu. 
4. Fourth incarnation was Nar-Narayan. Nar and Narayana is the twin sage incarnation of Lord Vishnu. In this avatar, he was wearing a jata on his forehead. The swan in his hands, the chakra at the feet and the thorax: in the place where the signs of Shrivatsa. He was born on earth to establish truth, justice, righteousness and other elements of religion. 
5. Fifth incarnation was Kapil muni. His father's name was Maharishi Kardam and mother's name was Devahuti. Kapil muni propounded Sankhya philosophy. 
6. Sixth incarnation was Dattatreya. As per story, once Goddess Lakshmi, Parvati and Saraswati became extremely proud of their ancestors. God created Leela to destroy their ego. According to him, one day Naradji reached Devlok while wandering around and told the three ladies in turn that your saintliness was nothing in front of Anusiya, the wife of sage Atri. The three goddesses told this thing to their husbands and asked them to test the rituals of Anusia. Then Lord Shankar, Vishnu and Brahma came to the ashram of Atri Muni dressed as a monk. Maharishi Atri was not in the ashram at that time. The three asked for alms to the goddess Anusiya, but also said that you have to give alms to us without getting naked. At first Anusiya was shocked to hear this, but then fearing that the sadhus should not be insulted, she remembered her husband and said that if my religion is true, then these three sadhus should become children of six months. As soon as he said this, Tridev started crying as an infant. Then Anusuiya became a mother and breastfed them with her arms and started swinging in the cradle. When the three gods did not return to their places, the ladies were distraught. Then Narada came there and told the whole thing. The three goddesses came to Anusaiya and apologized. Then the goddess Anusuiya made Trideva in her former form. Pleased, Tridev gave him a boon that all three of us would be born as sons from your womb. Then Moon was born from the part of Brahma, Durvasa from the part of Shankara and Dattatreya from the part of Vishnu.
7. Seventh incarnation was Yagya or yajna. Yagya was born in Swayambhuva Manvantara to protect it. As per story, Akuti was born from the womb of Shatrupa, the wife of Swayambhuva Manu. 
8. Eighth incarnation was Rishabhdev. As per story, Maharaja Naveli had no children. For this reason he performed a yajna with his wife Merudevi, wishing for a son. Pleased with the sacrifice, Lord Vishnu himself appeared and gave a boon to Maharaj Nabhi that I will be born here as a son. After some time, Lord Vishnu Maharaj was born as a boon in the form of a son. Seeing the son's exceedingly well-structured body, fame, oil, strength, opulence, fame, might and valor, etc., Maharaj Nabhi named him Rishabh (superior).
9. Ninth incarnation was Adiraj Prithu. As per story, Prajapati named Anga was married to Sunitha. He had a son named Wayne. He refused to believe in God and asked to worship himself. Then the Maharishi slaughtered him with the mantra pot. Then the Maharishi churned the arms of the sonless King Ven, from whom a son named Prithu was born. Seeing the lotus sign in the circle and feet in the right hand of Prithu, the sages told that the fraction of Shrihari himself has descended in the guise of Prithu.
10. Tenth incarnation was matsyavatar (fish) to save the world from destruction. As per story, King Satyavrat was giving Jalanjali a bath in the river one day. Suddenly a small fish came to his Anjali. When he thought to put it back in the ocean, but that fish said - you don't put me in the ocean, otherwise big fish will eat me. Then King Satyavrat kept the fish in his kamandal. When the fish became bigger, the king kept it in his lake, and then on seeing it, the fish became bigger. The king understood that this is not an ordinary creature. The king pleaded with the fish to come in real form. Hearing the king's prayer, the lord Vishnu appeared, and said that this is my mermaid. God told Satyavrat - listen King Satyavrat! There will be a catastrophe seven days from today. Then a huge boat will come to you with my inspiration. Take the subtle body of sapta sages, medicines, seeds and creatures and sit in it, when your boat starts to waver, then I will come to you as a fish. At that time you tie that boat with my horn by Vasuki Nag. At that time, I will answer you by asking questions, so that my glory which is known by the name of Parabrahma will be revealed in your heart. Then, when the time came, the fishery god Vishnu preached philosophy to King Satyavrat, who is famous by the name Matsyapuran.
11. Eleventh incarnation was Kurma or Kachhap (turtle). As per story, once Maharishi Durvasa cursed Indra, the king of the gods, to dehumanize him. When Indra went to Lord Vishnu, he asked him to churn the ocean. Then Indra agreed to churn the ocean together with the demons and gods. To churn the sea, the Mandarachal Mountain was made a churner and Nagraj Vasuki was netted. The gods and demons, forgetting their differences, uprooted Mandarachal and took him towards the sea, but they could not take him far. Then Lord Vishnu placed Mandarachal on the beach. The gods and demons made Mandarachal into the sea and made Nagaraja Vasuki a leader. But due to no base below Mandarachal, he started drowning in the sea. Seeing this, Lord Vishnu took the form of a giant kurma (turtle) and became the basis of Mandarachal in the sea. Mandarachal started moving fast on the huge back of Lord Kurma and thus the sea churning was completed.
12. Twelfth incarnation was Dhanvantri. He is also considered as the master of medicines. As per story, when the gods and the demons together churned the ocean, the first poisonous poison came out of it which was consumed by Lord Shiva. After this, uchhashrava horse, Goddess Lakshmi, Airavat elephant, Kalpa tree, Apsaras and many other gems came out of the ocean churning. At the end, Lord Dhanvantari appeared with the nectar urn.
13. Thirteenth incarnation was Mohini. As per story, during samundra manthan, asuras and devas started fighting for amrit (nectar).  Then Lord Vishnu incarnated Mohini to fascinate everyone. Mohini said, I will make devas and demons drink the nectar. Both agreed. The gods sat on one side and the demons on the other side. Then Lord Vishnu, who was in the form of Mohini, started singing and distributing nectar to the gods and demons. In reality, amrit was drinking only devas, while the demons were thinking that they were also drinking nectar. 
14. Fourteenth incarnation was Narsingh (half-lion man). Lord Vishnu killed Hiranyakashipu, the king of demons, with the incarnation of Narsingh. As per story, Hiranyakashipu, considered himself stronger than God. He had a boon that he would not die from man, god, bird, animal, neither in day, nor at night, nor on earth, nor in the sky, nor by weapon, nor by weapon. His son's name was Prahlad. Prahlada was an ardent devotee of Lord Vishnu since childhood. When Hiranyakashipu came to know about this, he became very angry and tried to convince Prahlada, but even when Prahlada did not believe Hiranyakashipu gave him the death penalty. Every time he survived. Hiranyakashipu's sister Holika, who had the boon of not burning with fire, sat in a blazing fire with Prahlada. Even then, by the grace of Lord Vishnu Prahlada survived and Holika was burnt. When Hiranyakashipu was about to kill Prahlada himself, Lord Vishnu appeared from the pillar with the incarnation of Narsingh and killed Hiranyakashipu with his nails.
15. Fifteenth incarnation was Vamana. As per story, Prahlada's grandson Daityaraj Bali took control of heaven. All the gods went to Lord Vishnu to avoid this calamity. Then Lord Vishnu said that I myself will be born from the womb of Devmata Aditi and will give you the kingdom of heaven. After some time Lord Vishnu incarnated Vamana. Once when Bali was performing a great yajna, Lord Vamana went to the sacrificial fire and asked for three feet of land from King Bali. King Bali's guru Shukracharya understood the lila of God and asked Bali not to donate. But Bali still pledged to donate three feet of land to Lord Vamana. Lord Vamana took a huge form and measured the earth in one step and the heaven in the second step. When there was no place left for the third step, Bali asked Lord Vamana to keep the step on his head. He reached Suthalok by placing a foot on Bali's head. Seeing the greatness of the sacrifice, God also made him the lord of Suthaloka. In this way Lord Vamana helped the gods and returned them to heaven.
16. Sixteenth incarnation was Hayagreeva. As per story, once two powerful demons named Madhu and Kaitabh reached the abyss after killing Vedas from Brahma. After the demise of the Vedas, Brahmaji became very sad and reached Lord Vishnu. Then God incarnated Hayagreeva. In this incarnation the neck and face of Lord Vishnu was like a horse. Then Lord Hayagreeva reached the abyss and killed Madhu-Catabh and gave Vedas again to Lord Brahma.
17. Seventeenth incarnation was Shri Hari. As per story, a powerful Gajendra lived with his arms in the valley of the mountain called Trikuta. Once he went to bathe in the pond with his arms. There a crocodile grabbed his leg and started pulling it under water. The struggle of Gajendra and crocodile continued for a thousand years. In the end Gajendra collapses and he meditates on Lord Srihari. Hearing the praise of Gajendra, Lord Srihari appeared and killed the crocodile with his chakra. Lord Srihari saved Gajendra and made him his councilor.
18. Eighteenth incarnation was Parshurama. As per story, the city of Mahishmati was ruled by the powerful Haiyavanshi Kshatriya Kartavirya Arjuna (Sahastrabahu). He was very arrogant and also tyrannical. Once Agnidev requested him to take food. Then Sahastrabahu came in boasting and said that you can get food from wherever you want, everywhere I rule. Then Agnidev started burning forests. In a forest, the sage Aapava was doing penance. Agni also burnt his ashram. Enraged by this, the sage cursed Sahastrabahu that Lord Vishnu would be born as Parashurama and destroys not only Sahastrabahu but all the Kshatriyas. Thus Lord Vishnu was born as the fifth son of Maharishi Jamadagri in the Bhargava clan.
19. Ninteenth incarnation was Maharishi Ved Vyas. He appeared as the son of the great sage Maharishi Parashar.
20. Twentieth incarnation was Hans. Once Lord Brahma was sitting in his assembly. Then his psyche son Sanakadi reached there and started discussing about the salvation of humans with Lord Brahma. Then Lord Vishnu appeared in the form of Mahahans and he removed the doubts of Sunkadi munis.
21. Twenty-first incarnation was Shree Rama. In Tretayug, Raksharaj Ravana had a lot of terror. For his slaughter, Lord Vishnu was born as a son from the womb of Mother Kaushalya at King Dasharatha. In this incarnation, Lord Vishnu killed many demons and lived his life by following Maryada. He went to exile at the behest of father. While suffering exile, Raksharaj Raja took away his wife Sita. In search of Sita, Lord reached Lanka, where there was a fierce battle between Lord Rama and Ravana in which Ravana was killed. 
22. Twenty-second incarnation was Shri Krishna. In Dwaparyug, Lord Vishnu took the Shri Krishna avatar killed Kansa. In the war of Mahabharata, became the charioteer of Arjuna and gave the knowledge of Gita to the world. 
23. Twenty-third incarnation was Lord Buddha He established Buddhism.
24. Twenty-fourth incarnation would be Kalki. According to religious texts, Lord Vishnu will incarnate as Kalki in Kali Yuga. Lord Kalki will be born as a son of an ascetic Brahmin named Vishnusisha at a place called Shambhal in Moradabad district of Uttar Pradesh. Kalki, riding a horse named Devadatta, will destroy the sinners from the world and re-establish religion.
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    Source: 
    1. Shri Ramchandra Keshav Dongra ji Maharaj
    2. https://www.ajabgjab.com/24-avatars-incarnations-of-lord-vishnu.html