In Hindu mythology God is depicted as Trinity, i.e. Brahma - The Creator, Vishnu - The Preservator & Mahesh or Shiva - The Destroyer. God in the form of Brahma does creation at the start of this world, God in the form of Vishnu does preservation or maintenance of this world, while Shiva finally annihilates this whole world at the end. So in this aspect while Lord Vishnu is responsible for the preservation or maintenance of this world, so He comes on this earth from His celestial abode (Vaikuntha) from time to time for the establishment of Dharma (righteousness). In such a role He puts law & order in it's place on earth & kills the evil elements in the society. So such a coming on this earth of Lord Vishnu is called descending of an Avatar of Lord Vishnu.
Avatar or Avatara (अवतार, Avatāra), is incarnation of God on earth. It literally means descent (avatarati) and usually implies a deliberate descent from higher spiritual realms to lower realms of existence for special purposes. Lord Sri Krishna was one of those Avatars of Lord Vishnu. In the past there had been so many Avatars of Lord Vishnu. In Lord Sri Krishna's case you can call these as His previous births but with a special purpose to establish Dharma (righteousness) on this earth. Below are verses from Bhagwat Gita which explain that only:
"yada yada hi dharmasya
glanir bhavati bharata
tadatmanam srjamy aham" (Bhagwat Gita: Chapter Four verse 7)
vinasaya cha duskritam
sambhavami yuge yuge" (Bhagwat Gita: Chapter Four verse 8)
"Sri Krishna said to Arjuna: Whenever Dharma, or the situation of law and order, is endangered on this world, I incarnate onto this world to re establish Dharma, law and order, and to protect the Sadhus or saints and to destroy the evil elements of the society. I do so appear, millennium after millennium."
So here below are 10 Avatars of Lord Vishnu in which He had descended on earth (In Kalki Avatar's case shall descend) in different forms to establish Dharma (righteousness) & Lord Sri Krishna was one of these Avatars.
1) Matsya Avatar (Fish): Lord Vishnu appeared in this form during "Satya Yuga". In this Avatar He had taken the form of a fish during a deluge that submerged the earth. Lord Vishnu commanded a 'Rishi' to gather together all samples of species and wait in a boat. The gigantic golden fish then dragged the boat through the deluge and then enabled Bhrama to start the act of creation all over again. Yet another legend has it that a demon once stole the four Vedas (Hindu religious texts) and hid them under the sea. Lord Vishnu assumed the Matsya (Fish) form and retrieved them and then restored them to their original shape. This legend is held at the Parimalaranganathar temple at Mayiladuturai, near Thanjavur in Tamilnadu state of India.
2) Kurma Avatar (Tortoise): Lord Vishnu appeared in this form during "Satya Yuga" only. In KURMA Avatar, Lord Vishnu incarnated himself as a tortoise & helped in churning of the ocean of milk after adding medicines into the ocean (Samudra Manthan) to get the nectar for the gods (Devtas).
3) Varaha Avatar (Boar): Lord Vishnu appeared in this form during "Satya Yuga" only. Vishnu took the form of a huge wild boar - the Varaaha Avatara, dived into the ocean, and saved Bhoomi Devi (Earth) who had sunk into the bottom of the ocean. Vishnu as Varahamurthy is enshrined at Tiruvidandai near Chennai in Tamilnadu state of India, and also at the Kanchipuram Kamakshiamman temple.
4) Narasimha Avatar ( Half lion half human): Lord Vishnu appeared in this form during "Satya Yuga" only. In this Avatar, Vishnu destroyed the demon king Hiranyakashipu and demonstrated his omnipresence in a powerful way.
5) Vamana Avatar (the dwarf or midget): Lord Vishnu appeared in this form during "Treta Yuga". In this form Vishnu took the form of Vamana the midget to destroy the demon Bali. He visited Bali during a sacrifice where the latter was arrogantly distributing gifts of the seeker's choice, to show his power of wealth. Vamana meekly asked for three feet of land measured by his small feet to which Bali agreed. All on a sudden Vamana the midget assumed the massive form of Trivikrama, dominating the whole of the universe. With his first foot He covered the earth, with the second the heavens. When there was no room for the third, Bali, who never went back on his word offered his head, and Vishnu sent him to the netherworld. Trivikrama is enshrined at Tirukkovilur, kanchipuram and Sirkazhi in Tamilnadu state of India.
6) Parasurama Avatar (Rama with the axe): Lord Vishnu appeared in this form during "Treta Yuga" only. In this form Vishnu quelled the arrogance of the Kshatriya (warriors) rulers who harmed the sages and unprotected mortals.
7) Rama Avatar (king of Ayodhya): Lord Vishnu appeared in this form during "Treta Yuga" only. In this form Vishnu appeared as Ramachandra, the prince and king of Ayodhya who killed demon king Ravana. The great epic Ramayana speaks of the glory of the Rama Avatar of Lord Vishnu.
8) Krishna Avatar (meaning "dark coloured" or "all attractive"): Lord Vishnu appeared in this form during "Dwapara Yuga along with his brother Balarama" Mahabharata speaks of Balarama and Krishna in detail.
9) Balarama Avatar: Some of the texts call Balarama only as the ninth Avatar of Lord Vishnu, while many others call "Gautama Buddha" as the ninth Avatar of Lord Vishnu. Such believers say that Lord Vishnu took up the Avatar of Buddha to purify Hinduism of excessive ritualism. He preached detachment, and the middle path consisting of eight fold virtues of right views, right resolve, right speech, right conduct, right livelihood, right effort, right mindfulness and right meditation.
10) Kalki Avatar ("Eternity", or "time", or "The Destroyer of foulness"): Lord Vishnu is expected to appear in future in Kalki Avatar at the end of Kali Yuga, the time period in which we currently exist. It is believed that at the end of the current Kali Yuga, there will be a deluge when Kalki - the tenth and the last Avatar of Vishnu, will ride forth on a horse to redeem humankind and re-establish righteousness.