Lord Vishnu is the Preserver or Sustainer of life with his steadfast principles of order, righteousness and truth. WHEN THESE VALUES ARE UNDER THREAT, VISHNU EMERGES OUT OF HIS TRANSCENDENCE TO RESTORE PEACE AND ORDER ON EARTH.
His Ten Avatars are
Matsya (fish)
Koorma (tortoise)
Varaha (boar)
Narasimha (half man half lion)
Vamana (the dwarf)
Parasurama (the angry man)
Sri Rama (the perfect human)
Balarama (Sri Krishna's brother)
Sri Krishna (the divine diplomat & statesman)
Kalki (yet to appear 10th Avatar).

MATSYA - 1st of Lord Vishnu's Dasavathar.

Once King Manu was under strict penance and was performing ablutions in the water of River Kritamala (Vaigai). He took a handful of water and was surprised to find a small fish which requested him for rescue. When this small fish started growing extraordinarily in size, the king realised that it should be Lord Vishnu in the form of fish. At once the king prostrated and the God predicted that the world will come to an end shortly, ordered the king to carry each species in a huge boat which HE appearing as a gigantic fish would propel the boat to Mount Himavat. Everything went according to the prediction and thus King Manu, being a noble man and an ardent devotee could become successful in this allotted work.

KOORMA- 2nd of Lord Vishnu's Dasavathar.

Once Sage Durvasa presented a flower garland to Lord Indra who just gave it to his elephant Airavatam which trampled the garland. The angered sage cursed the Devas to lose their immortality. In order to regain their lost immortality, the Devas approached Lord Vishnu to help them in their great task of churning the Milky Ocean to extract Amrutam (Nectar) from the depths of the ocean. The divine serpent Vasuki was used as a rope for churning the Mandara mountain which was placed on the hard shell back of the Koorma Roopam of Lord Vishnu.
Thus Lord Vishnu incarnated taking the form of amphibian (living both in land and water) which is a gradual evolution to higher form than the previous Pisces form (totally aquatic).

VARAHA - 3rd of Lord Vishnu's Dasavathar

During ancient time, a powerful Asura (Demon) named Hiranyaksha was creating havoc in the three worlds. Once while Lord Brahma was in deep meditation, Hiranyaksha snatched Mother Earth (Bhoodevi), the four Vedas (Rig, Yajur, Sama and Atharvana) and other sacred scriptures. He pushed all these into the depths of the ocean. In order to re-position Mother Earth, Lord Vishnu had to incarnate in the form of Varaha (wild boar) to dive into the deep ocean. After a fierce fight with the Asura, Lord Vishnu in the form of wild boar killed the demon with HIS sharp tusks, got Mother Earth back and re-positioned her. Thus from the previous amphibian form, there is an evolution to higher form of a mammal/beast.

NARASIMHA-4th of Lord Vishnu's Dasavathar

A mighty demon king Hiranyakashipu gets a boon from Lord Brahma that his defeat and death is highly impossible and neither a man or animal can kill him. Prahlada, son of Hiranyakashipu is a staunch devotee of Hari (Lord Vishnu). The father feels his son is his enemy and tries to kill him but in vain since everytime Prahlada was attacked, the God would save him. In a fit of anger Hiranyakashipu asks his son who is this Hari and where does he live, the son answers that the Lord is all pervading and is present everywhere. Pointing out to a nearby pillar, the father asks whether Hari is here, Prahlada says 'yes'. Now the outrageous demon king hits the pillar with his mace, it slits into two halves and from within comes the strange being with Lion's head and human body roaring and pouncing on Hiranyakashipu and kills him. This avatar signifies the gradual evolution of a wild beast to a beastly human form.

VAMANA-5th of Lord Vishnu's Dasavathar

King Bali, the grandson of Prahlada was a great devotee of Lord Vishnu, conquered the three worlds. Due to this Indra and other devatas feared their positions and sought help from Lord Vishnu who took the form of Vamana (a dwarf ascetic). Thus Vamana appeared before King Bali and requested for three feet of land which the King accepted gladly though he was warned by his Guru - Sukracharya. Lord Vishnu who was in the form of a dwarf took a gigantic form and covered the entire Earth with one foot, the entire heaven or sky with the second foot. When Lord Vamana asked for the third foot, King Bali, who was fully aware that it was Lord Vishnu asked to place the third foot on his head. Thus, King Bali was pushed to the Patala loka or the netherworld. In this avatar, the evolution stage is from beastly man to a dwarf human where there is a gradual evolution of civilized creature.

PARASURAMA - 6th of Lord Vishnu's Dasavathar

The incarnation of Lord Vishnu as a Brahmin who was born with unusual physical strength to Sage Jamadagni and Renuka. He was gifted with a Parasu or Axe by Lord Shiva and thus always holds it in his hand. He takes revenge against kshetriyas as his father was killed by the sons of King Kartavirya who steals Kamadhenu (the gifted cow) from Jamadagni. In this incarnation, there is a further evolution to a powerful human from a dwarf man in the previous avatar.

SRI RAMA - 7th of Lord Vishnu's Dasavathar

Raguvamsam by Brahmasri Anantharama Deekshitar (3min Audio)
Sri Rama, the eldest son of King Dasaratha and Queen Kausalya in Ayodhya, is one of the most important avatars of Lord Vishnu in the Treta Yuga. The divine character of Sri Rama, represents the ultimate embodiment of truth, of morality, of chivalry and virtue. An ideal son, an ideal husband, an ideal brother and above all, an ideal king - Sri Rama is referred as MARYADHA PURUSHOTTAMA – the Perfect Man/Lord of Self-Control/Lord of Virtue. Lord Vishnu took this avatar to combat Ravana - the most powerful demon king. Considering the stages of evolution from simple to complex beings, from a powerful enraged human there is a gradual transformation to highly civilized human.

BALARAMA-8th of Lord Vishnu's Dasavathar

Lord Vishnu incarnated as Balarama when Indra and the other gods came and asked for release from the demon son Kamsa, king of Mathura. Vishnu took one black and one white hair from his head and promised that these two would come to fight against the demons. This way, first Balarama (the white hair) and then Krishna (the black hair) incarnated in the womb of Devaki. To save him from Kamsa, Balarama was transferred secretly from Devakis womb to that of Rohini´s, from which he was born. Both the brothers Balarama and Krishna grew up in a cowherd's family.
At an early age, he killed the demon Dhenuka, accompanied Krishna to Mathura and supported him until Kamsa was killed. Balarama also released Krishna´s son, Samba, from Duryodhana's prison.
Balarama is a symbol of strength, an obedient son, an ideal brother and husband. He is a protector from desires, symbolizing duty, honesty and simplicity. He increases the transcendental bliss of others, hence the name 'Rama', because of his extraordinary strength he is also called as 'Balabhadra'.
SRI KRISHNA-9th of Lord Vishnu's Dasavathar
Mother Earth and Lord Brahma unable to bear the burden of sins committed by evil kings and rulers like Kamsa, prayed to the Lord Vishnu, who assured them that he would soon be born as Sri Krishna to end these forces. Kamsa, the ruler of Mathura (in northern India) utterly terrified his people by his actions. On the day of his sister Devaki's marriage, an akashvani or voice from the sky was heard prophesying that her 8th son would be his destroyer, frightened by this Kamsa imprisoned both Devaki and her husband Vasudeva thereby killing all the new borns until the 8th one. Devaki and Vasudeva prayed to Lord Vishnu, who appeared before them and said he himself was coming to rescue them and the people of Mathura. Vasudeva was told to carry the Lord just after birth to the house of his friend, the cowherd chief Nanda in Gokula, where Nanda's wife Yashoda had given birth to a daughter, exchange his boy and bring Yashoda's baby daughter back to the prison.
At midnight on ashtami , the divine baby was born in Kamsa's prison. Remembering the divine instructions, Vasudeva held the baby above his head (protected by five-mouthed snake-Adisesha) and reached Gokula after crossing river Yamuna (which gave way). He exchanged the babies and hurried back to the prison of Kamsa with the baby girl. Early in the morning, all the people at Gokula rejoiced the birth of Nanda's beautiful male child. When Kamsa came to know about the birth, he rushed inside the prison and tried to kill the baby. But this time it skipped from his hand and reaching the sky. She was transformed into the goddess Yogamaya, who told Kamsa: "O foolish! What will you get by killing me? Your nemesis is already born somewhere else."
In his youth Krishna killed Kamsa along with all his cruel associates, liberated his parents from prison, and reinstated Ugrasen as the King of Mathura.
Lord Krishna preached Srimad Bhagavath Geeta to Prince Arjuna in the battlefield of Mahabharat who was emotionally affected fighting his own kith and kins.
This incarnation proves that there is an evolution of higher human with high intelligence and smart human who could face all the difficulties right from childhood and thus achieve success.

KALKI-10th of Lord Vishnu's Dasavathar

Kalki avatar is yet to be incarnated. It is believed that Lord Vishnu will incarnate himself as Kalki, riding a white horse with blazing sword in his hands. At the end of Kali Yuga (present eon), he will punish all evil doers in this world, destroy this world and recreate a golden age again.
As Lord Sri Krishna says in Srimad Bhagavad Gita - Ch 4 Slokas 7 & 8
O Bharata, whenever there is decline of righteousness and rise of evil, I manifest Myself.
For the protection of the righteous, for the destruction of wicked, and for the establishment of Dharma, I am born in every age.