Who is an Avataar
An Avataar represents the direct descent of God Himself to the human level in order to help the human beings to ascend to the divine level, of which the liberated soul is a perfect example. A Jeev is forced to take a body due to his past actions whereas the Avataar incarnates of His free will for helping mankind as He is God Himself. Consequently, He is always conscious of his mission and power. It is only He who can see both the Absolute (Brahm) and the manifested world simultaneously, can teach the world about the Absolute. Vedaant admits the God's manifestation in the world, his omniscience as also omnipotence and grace. Hence there is nothing unreasonable in the doctrine that he can also assume a special and unique form as an Avataar. God incarnates Himself in response to the needs of the times. So, there are no restrictions regarding either the number of incarnations or the place. Bhaagavat Puraan gives three lists of Avataar - one list of 22 names (1.3.6-22); second ,list of 23 names (2.7.1) and third list of 16 names (11.4.3.) and goes to the extent of declaring that the Avataar are innumerable (1.3.26). This is true for Vishnu only. There is a belief that Shiv took 100 Avataar among Shaivites.

Types of Avataar
Two types of Avataar are said - Poorn (complete) Avataar and Ansh (part) Avataar. When Bhagavaan takes Avataar with all His powers - 16 Kalaa, He is called Poorn Avataar; and when He comes with His limited powers, that Avataar is called His Ansh Avataar. Among all the Avataar, His Krishn Avataar is said to be His Poorn Avataar. It means that Krishn was He, the God Himself, with all of His powers. His all other Avataar are said to be the Ansh Avataar.

[Taken from Webonautics]
Among 16 Kalaa, in a Poorn Avataar,
5 Kalaa are shared with animals - 5 Gyaan Indriyaan (smell, seeing, hearing, touch and taste) and
4 Kalaa with human beings - mind, heart, intelligence and Tureeya Avasthaa (trnscendence of intutive experience)

The 7 phases that follow are the characteristics of Poorn Avataar -
(1) Grace or reward for effort that fails to be rewarded though it has come from the deserving
(2) Anugraha or special grace whether the recipient merits it or not
(3) The power to create a new order of life in society, new status of consciousness in individual or new objects
(4) Power to support and sustain what is inherently good which may happen to be defenseless
(5) The power to destroy evil
(6) The assumption of a form which, whenever it is recalled mentally, or in the presence of the Avataar Himself, affording a solution that the beholder has in mind; and
(7) The assumption of a name which has a similar potency.

Vishnu's Avataar :
The Shlok 6.11 of Bhagavad Geetaa : -.
Eko Devaha sarvabhooteshu goodhahasarvavyaape e sarvabhootaantaraat maakarmaadhyaksh
aha sarvabhootaadhivaas ahasaakshee chetaaha kevalo nirgunashcha
Vishnu literally means one who pervades everywhere. He is predominantly Saatwik and is responsible for preservation.
He takes incarnations to protect Dharm and to punish A-Dharm, - Matsya (fish), Koorm (tortoise), Varaah (boar), Narasinh (man lion), Vaaman (dwarf), Parashuraam, Shree Raam, Shree Krishn, Buddha and finally Kalki (yet to come).

Avataar means incarnation of God. Raam, Krishn, Mohinee all were Avataar. There are three types of bodies (see also Types of Body) - Empirical Body - that is our physical body which we can touch; Another Body which we see in dreams - we can see it, we can feel by that body but we cannot touch it; thus although it is a body but still it is not a body; and the third body is Avataar's body - they come for some purpose and go away after fulfilling that purpose. These bodies have no mind. Because we need mind to get knowledge. Since Avataar don't need any kind of knowledge, because they already have it, that is why they don't need this mind, so they don't have it. Otherwise they are like human beings but certainly with special powers of God. Avataar possesses all characteristics of God.

This is also in consonance with the Darwinian theory of evolution also!!
(1) Matsya (Aquatic),
(2) Koorm (both aquatic and on earth),
(3) Varaah (on earth only),
(4) Narasinh (half man, half beast)..
(5) Vaaman (dwarf man),
(6) Parashurama( balancer),
(7) Raam (Purushottam - idealist),
(8) Krishn (Poornaavataar),
(9) Buddha (reviver from anarchy) and
(10) Kalki (yet to come).

Avataar are classified into three types.- Poornavataar, Anshavataar, and Aaveshaavataar. Great sage Ved Vyaas may be considered as Anshaavataar. When there is a temporary entry of the divine as in the case of Narasinh manifesting through Padmapaad (a disciple of Shankar) it is called Aaveshaavataar.

The fact that Shankar did not forbid us from worshipping God through the various names and shapes - in fact he was responsible to construct and restore several temples including the famous Badaree Naaraayan Temple and the Shree Shail Mallikaarjun Temple etc. His compositions of beautiful poems for prayer offered to almost all deities of Sanaatan Dharm shows that there was a balance in his teachings. He did not advocate A-Dwait for everyone and without preparations. In fact his concept was, "Prahamaa Pratimaa Poojaa, Jap Stotraani Madhyam, Uttamaa Maanasee Poojaa, Soham Poojottamottam.