DASHAVATARA

Vishnu has had many incarnations with time, however, the Dashavatara refer to his ten primary (i.e. full or complete) avatars. He took these forms on earth to restore the cosmic order. This has been described in the following shloka-
यदा यदा हि धर्मस्य ग्लानिर्भवति भारत|
अभ्युत्थानंधर्मस्य तदात्मानं स्रुजाम्यहम्|
परित्राणाय साधूनां विनाशाय च दुष्कृताम्|
धर्मसंस्थापनार्थाय सम्भवामि युगे युगे|
Which translates to –
“Whenever there is a decay of righteousness, O Bharata,
And there is an exaltation of unrighteousness, then I descend myself,
For the protection of the good, for the destruction of evil-doers,
For the sake of firmly establishing righteousness, I am born from age to age.”
The list includes the following Avatars or incarnations: Matsya, Kurma, Varaha, Narsimha, Vamana, Parashurama, Rama, Krishna, Balrama or Buddha and Kalki. Some traditions include a regional deity such as Vithoba or Jagannath in penultimate position, replacing Krishna or Buddha. It is interesting to note how many scholars believe that these incarnations actually represent the theory of Evolution.
However, it must be noted that Aurobindo regarded the “Avataric Evolutionism” as a “parable of evolution”, one which does not endorse evolutionism, but the “transformative phases of spiritual progress”. Some Vaishnavies also reject this “Avataric Evolutionism” concept. For example, ISKCON states that this theory degrades the divine status of Rama and Krishna, unduly sequences Rama as inferior to Krishna, and both to the Buddha. They believe that both Rama and Krishna are supremely divine, each right and perfect for the circumstances they appeared in.
INCARNATIONS

Matsya
Matsya or the fish was the first incarnation of Vishnu in the Satya Yuga. It is believed to be during the Paleozoic era or between 542 million to 251 million years ago when the breakup of one supercontinent to form another (Pangea) took place. Since lifeforms were not very evolved at that time, the Matsya is depicted as a giant fish or a human torso connected to the rear half of a fish, like a mermaid.
The story goes that Vishnu in the form of Matsya had warned the first man, Vaivasvata Manu (often called Manu), of a great flood which would end the three worlds. Matsya asked Manu to bring one of every plant and animal species to the shore and on the day of the great flood he safely took all of them to a new world in a boat as that the fish pulled it, thus, saving humanity and all the other animals.

Kurma
Kurma or the giant tortoise was the second incarnation that came about in the Satya Yuga, supposedly belonging to the Mesozoic era, an interval of time from about 252 to 66 million years ago. It is depicted as a giant tortoise or a mixed form of human and tortoise.
The story goes that there was a war between the Gods and Demons during the Churning of the Ocean of Milk or the Sagarmanthan to procure the elixir of immortality or amrit. Unfortunately, Mount Mandara (the mountain used as the churning staff) started to sink, so Vishnu took the form of a giant tortoise to bear the weight of the mountain on his back. Despite the efforts of the gods, the demons took the nectar and so Vishnu had to disguise himself as Mohini and take the nectar from the demons.

Varaha
Varaha or the boar appeared to defeat Hiranyakasha who stole the Earth (personified as Bhudevi) and took it to the bottom of the cosmic ocean or primordial waters causing serious destruction that takes place at the end of each age. The battle between Varaha and Hiranyaksha is believed to have lasted for a thousand years, which the former finally won by slewing the demon. He then carried the Earth out of the ocean between his tusks and restored it to its place in the universe.
Vishnu in this incarnation is represented in the form of a half man and half boar or a full boar. This incident took place in the Satya Yuga, corresponding to the Cenozoic era in history, where herbivorous animals evolved and non-avian dinosaurs dissapeared leading to a rise of mankind. It spans from only about 65 million years ago.

Narasimha
Narsimha was Vishnu’s last incarnation in the Satya Yuga and represents the last half human/half animal incarnation of Vishnu. It represents the time in the Cenozoic era when the carnivorous animals evolved. In this Avatar Vishnu took the form of a man with the head and claws of a lion.
Hiranyakashyap was a demon with a boon from Brahma that he could not be killed by any man or animal, inside or outside a room, on ground or in air, during the day or night, on earth or the stars, with a weapon that was living or inanimate. Because of his boon and supposed invincibility, Hiranyakashyapa started creating havoc by not letting anyone preach any religion, persecuting anyone who dared to disobey him, including his own son, Prahlada. However, Prahlad’s immense dedication led to Vishnu taking the form of Narasimha, who disemboweled Hiranyakashyap at the threshold of his house at dusk, with claws, in his thighs, giving the the subjects of the king the right to preach any religion.

Vamana
The fifth incarnation of Vishnu, the Vamana was his first incarnation of the Treta Yuga. Here he came to earth in the form of a dwarf, so this phase represents growing dwarfs and the first steps towards the human form ‘from monkeys’.
In the iconography, Vamana as a dwarf can be seen carrying a wooden umbrella.
The legend that goes with him is that, Bali, the grandson of Prahlad was able to defeat Indra, the king of gods through his devotion and penance. This victory made him the supreme authority over the three worlds. The change of power frightened the gods and they went to Vishnu for protection who came to Earth and restoring the authority of Indra as the dwarf. He approached Bali during one of his Yajnas, and the king promise that he would fulfil one of the Dwarf’s wish. So the Vamana asked for three paces of land. Bali didn’t retract from his word even after being warned by his guru. The dwarf then transformed into his giant Trivikrama form (tri or ‘three’ + vikrama ‘step’ or ‘stride’) becoming ever larger than the Earth. Vamana with one step covered the whole earth, and with the second step the middle world between earth and heaven. As there was nowhere left to go, King Bali was unable to fulfil his promise, so he lowered his head and suggested that the Vamana place his foot on it for the promised third step. Vamana then placed his foot on the king to push him in hell. Yet, due to the king’s humility Vishnu granted Bali immortality, making him the ruler of Patala, the netherworld. This legend appears in hymn 1.154 of the Rigveda and other Vedic as well as Puranic texts.

Parashurama
Parashurama is a hero with an axe, but, he is considered to be an imperfect human form from the Treta Yuga. He is the son of Jamadangi (one of the Saptarishis) and Renuka. He was the first Brahmin-Kshatriya (Warior-Sage) who got the axe after penance to Shiva. The axe represents the advent of the iron age.
The story goes that King Kartavirya Arjuna and his hunting party halted at the ashrama of Jamadagni, father of Parashurama. The saint fed the king and his army with the help of his divine cow Kamadhenu. The king was extremely pleased by the cow and demanded it from the saint who refused. The king was furious and took the Kamdhenu by force after destroying the Ashram. When Parashurama found out what had happened, he wanted to avenge the insult so he killed the king and his army singlehandedly with his axe. However, King Kartavirya’s son was out to seek revenge and killed Jamadangi. Since the prince was a Kshatriya, Parashurama vowed to kill every Kshatriya on earth twenty-one times over and filled five lakes with their blood as her mother had beaten her breast those many times in vain. It was only his grandfather Rucheeka, who was able to stop and calm him down. However, since Parashurama is a chiranjeevi or immortal he is believed to be alive even today doing penance at Mahendragiri.
Not only this, Parashurama is also credited for creating Kerala by throwing his mighty axe that landed at the sea, displacing water and creating a landmass that is now called Kerala.

Rama
Rama was the prince and then king of Ayodhya who is considered to be an ideal prince, without superpowers, despite being an incarnation of Vishnu. He came to Earth in the Treta Yuga. As a man with bow and arrow he represents people with developed skills using semi-modern weapons like bow and arrow.
Ramayana, or his story is one of the most widely read scriptures of Hinduism. He gave up his throne and went in to exile for fourteen years along with his brother Laxman and wife Sita to keep his father’s honour. While in exile his wife Sita gets abducted by the demon king of Lanka, Ravana who takes her to Ashoka Vatika in Lanka. so Rama travelled there to save his wife and kill the demon with the help of an army of monkeys. After which he goes back to Ayodhya and is crowned.

Krishna
Krishna was the eight Avatar of Vishnu and also the eight child Devaki and Vasudeva to kill his evil maternal uncle (the king) Kansa. Many believe that he came along another incarnation of Vishnu (Balarama), who is often regarded as the eight incarnation, Krishna being the ninth one, when Buddha is not included in the list.
Krishna was raised by his foster parents Yashoda and Nanda. He was cowherd, and also a philosopher and guide to the Pandavas (in the battle of Kurukshetra of Mahabharata) in the Dwapara Yuga thus, giving birth to the Bhagavad Geeta. He embodies several qualities such as love, duty, compassion, and playfulness. Usually depicted with a flute in hand, a peacock feather in hair along with a pitambar dhoti (or yellow dhoti). In fact, Krishna is often called the original Supreme Personality of Godhead, from whom everything else emanates.
In the evolution theory it is believed to represent the time when people began using superior weapons like his Chakra, started domestication and settled agriculture.

Buddha
Buddha or the enlightened one of the Dwapara Yuga is the founder of Buddhism who is often depicted in the “Hindu Scriptures” as a preacher who leads demons and heretics away from the path of Vedic scriptures or a compassionate teacher who preached non violence or ahinsa. He was included as an incarnation of Vishnu under Bhagavatism between 330 and 550 CE and this shows the advent or beginning of advancement towards the modern age.
The mythologies of the Buddha in the Theravada and of Vishnu share a number of structural and substantial similarities according to Indologist John Holt, who believes that Theravada states that Buddha covered earth to heaven and then placed his right foot over Yugandhara in three strides – a legend that parallels that of the Vishnu’s Vamana avatar. Similarly, Buddha, like Vishnu is claimed in the Theravada to have been born when dharma is in decline to preserve and uphold the dharma. These similarities, thus, may have contributed to the assimilation of the Buddha as an avatar of Vishnu.
By the 8th century CE, the Buddha was included as an avatar of Vishnu in Puranas, indicative of the Hindu ambivalence towards Buddhism. Conversely, Vishnu has also been assimilated into Sinhalese and Mahayana Buddhism in Buddha-Bhagavatism. By this period, the concept of Dashavatara was fully developed.
The story of Buddha goes that he was born as Prince Siddhartha who renounced the world to became a monk and attained enlightenment. Sacrificing the luxuries of princely life, detaching himself from worldly pleasures and practising deep meditation, he spread the message of peace. Vishnu came to earth in this form to make humans see the importance of self-realization and liberation however, some Buddhist scholars do not believe Buddha as the ninth incarnation of Lord Vishnu. They say it was a trick played by Brahmins to decrease the increasing influence of Buddhism when it was at its peak in India. Some Vaishnavites too don’t believe Buddha as the incarnation of Vishnu. Instead, they believe Balaram as 8th and Krishna as the 9th incarnation of Vishnu.

Kalki
Kalki is considered to be the final incarnation of Vishnu and is supposed to appear at the end of Kali Yuga. He will be on top of a white horse, with his sword drawn out blazing like a comet. He will appear when there is only chaos, evil and persecution, dharma has disappeared, so he will end the Kali Yuga to start again with Satya Yuga, thus, beginning another cycle of existence. He is therefore considered to be the saviour of mankind and destroyer of darkness and also the harbinger of the end of time. He is considered to be a very advanced form of a human being, both technologically and genetically.
SOURCES:
https://en.wikipedia.org/wiki/Dashavatara