Vamana- Fifth incarnation of Vishnu

According to Puranas, VISHNU incarnated in the form of  VAMANA to test demon king MAHABALI.. PRAHLAD had a grandson MAHABALI .PRAHLAD was a great devotee of VISHNU. He inculcated his teachings in MAHABALI. So MAHABALI also became a great devotee of VISHNU. MAHABALI became a wise,noble,righteous and generous king who ruled earth and heaven. Everyone under MAHABALI rule was happy and prospering. MAHABALI was a generous king and whosoever requested  him with anything was granted the same. Though generous MAHABALI became arrogant. Helping others is king duty but he forgot this and became very proud of it. So VISHNU incarnated in the form of a dwarf called VAMANA to test his generosity.

Vamana with Mahabali

One day after completing his morning prayers, he was preparing to donate to some poor people. Then VISHNU in the form of VAMANA arrived and requested for a piece of land.

Narsimha- Fourth incarnation of Vishnu

VISHNU took the form of  VARAHA to kill HIRANYAKSHA. After HIRANYAKSHA was killed his younger brother HIRANYAKASHIPU got very angry and planned to revenge his brother death from VISHNU. Like his brother HIRANYAKSHA, he worshiped BRAHMA with great devotion. He meditated for many years just to seek boon of immortality. Brahma was impressed with his devotion so he told HIRANYAKASHIPU to ask whatever he wants. As HIRANYAKASHIPU told BRAHMA that he wants the boon of immortality from him. BRAHMA told him that he cannot fulfill his wish. Then
HIRANYAKASHIPU asked BRAHMA to make his death nearly impossible. He asked that let no animal or human or god cause his death at day or at night with no weapon.Neither he can be killed inside or outside of any shelter nor on Earth or Sky and asked for the lordship of the whole world. Brahma granted his wish. So he became practically immortal. After getting his wish, he conquered the whole world and became a tyrant demon king. He banned the worship of Vishnu and declared himself as the sole god of the world. Followers of Vishnu were exploited and killed. Some followers seek the assistance from Vishnu. Lord Vishnu told them to be patient.

Narsimha kiiling Hiranyakashipu

INDRA the king of the gods attacked HIRANYAKASHIPU's palace in his absence.

Ancient Hindu technology by history channel

Varaha- Third incarnation of Vishnu

Ancient mythology says that after Kurma, Vishnu incarnated in the form of Varaha to protect Goddess Earth known as Bhudevi from the captivity of Hiranyaksha(Demon).According to the story Hiranyaksha captured Goddess Earth and took her deep into ocean. Goddess Earth cried for help and asked Vishnu to save her. Hiranyaksha was a meditator who tries to make Brahma(Lord of Creation) happy with his devotion. Impressed by his devotion Brahma blessed him that no animal or human can kill him. Varaha, a Boar head with  a human body which is neither an animal nor a human. Varaha went deep into underwater and fought Hiranyaksha. The battle lasted for many years and Hiranyaksha was killed in the end. Varaha rescued Goddess Earth from the torment of Hiranyaksha by placing her on his tusks and took Goddess Earth to her original position. Goddess Earth fall in love with Varaha and they married.

Varaha killing Hiranyaksha

Varaha is depicted as a form with Boar head and human body. Varaha original size was very small but soon he grew into a gigantic beast with stong built. His skin is dark in complexion and he carries a huge mace as his primary weapon.

Depiction of Varaha

Read this also KURMA

Solar System & Vedas

Solar System explained in Vedas

Read this also THE BEGINNING

Kurma-Second incarnation of Vishnu

After Matsya form, Vishnu incarnated in the form of Kurrma (tortoise form) so that churning of ocean takes place and nectar of immortality can be extracted for DEVAS (God). According to ancient Puranas, Durvasa was a sage who was bad tempered. Once he was wandering the earth, he saw a Nymph of air and she was wearing a floral wreath. Durvasa demanded floral wreath from Nymph. Nymph gave it to him respectfully. Wandering further Durvasa encountered Indra(king of God) who was riding on his Elephant. Durvasa throws floral wreath towards Indra as a blessing. Indra accepted it and put it on his Elephant. Irritated by the fragrance of floral wreath Elephant quickly throws the wreath on the ground. Seeing this Durvasa felt disrespected and got very angry so he cursed Indra that he and his God's powers will diminish and they will become weaker. Demon king Mahabali saw this opportunity and attacked the Gods. Gods flew away from Demons and seek the assistance from Vishnu. Vishnu told Gods to call a truce with Demons and prepare to extract the nectar of immortality from churning of the ocean. Gods did as they were told. Churning of ocean required a lot of strength and Gods were weak so they asked for help from Demons and in return they will share the nectar with the Demons. Demons agreed on this. But the ocean body was so large that it requires Mount Mandara as a churning pole and Serpent king Vasuki as a churning rope.
As God and Demons started churning the ocean, Mount Mandar begun to sink due to its own weight.Then Vishnu incarnated in the form of Kurma and supported the the Mount on his back from beneath. As the nectar of immortality emerged with other divine things,  Demons took the nectar and begun to quarrel among themselves. So Vishnu took a form of beautiful women which mesmerized the Demons with its beauty. Vishnu fooled the Demons and retrieve the nectar. Vishnu distributed the nectar among Gods. Gods regained there powers. As soon as Demon discovered this trick it was too late. God defeated the Demons with regained strength.

Depiction of Kurma

The Kurma form is often depicted as upper human half with lower half of a tortoise.

Churning of milk ocean

Read this also MATSYA

Matsya - First incarnation of Lord Vishnu

Lord Vishnu first incarnated into MATSYA, a fish form which Vishnu took to protect Manu along with seven Rishis from great flood. According to Puranas, Manu caught Matsya when he was a small fish. Matsya seeked protection from Manu who was endowed with great wisdom. So Manu decided to protect Matsya. Showing gratitude towards Manu, Matsya told him that he knows exactly when the great flood will occur. So he will protect Manu when the right time comes.Manu put Matsya in a jar to keep him safe. After some time Matsya had outgrown the jar. So Manu dug a pit amd pour some water into it and kept Matsya there for a long time. After Matsya outgrown the pit, Manu took him to the ocean and put him in it. Matsya told him to build a boat. He did exactly and built a boat. when great flood came the whole land sank in water. Manu called Matsya for help. Matsya towed Manu to highest and driest point on Earth protecting Manu's life

Matsya

Matsya is depicted with upper human half and a lower half that of a fish.
Read this also VISHNU