Avatar means to appear, to descend, to take birth or to manifest.
Avatar and Incarnation of the god considered as synonyms,
But there is a little difference if definition is considered. It requires the birth of God in physical form, through the natural process and his existence upon earth in physical form undergoing the same experiences as living beings. The following are the different types of Avatars of Supreme.


The one who don't have name and form [nirakara and nirguna] who have no qualities will take the form of his desire, which is known as Amsha [Expansion] Avatar.
For Example: Lord Bramha, as the birth of Lord Bramha is just by the desire of the Supreme Lord [Vishnu] [or, the Supreme God manifested himself as Lord Bramha to create the Universe].


Amshaamsha Avataras are those Avataras, which helps the purpose of the Amsha Avtharas
For Example: If Lord Bramha is Amsha Avathar and he has been created for the creation of the Universe and created Praja Pathis who helped further creation on earth.


In Bhagavat Gita, Lord Sri Krishna explained that he will descend himself whenever and wherever  there is a decline in religious practice [Dharma], and a predominance rise of irreligion [Adharma]

यदा यदा हि धर्मस्य ग्लानिर्भवति भारत । 
अभ्युत्थानमधर्मस्य तदात्मानं सृजाम्यहम्‌ ॥
परित्राणाय साधूनां विनाशाय च दुष्कृताम्‌ ।
धर्मसंस्थापनार्थाय सम्भवामि युगे युगे ॥

Yada Yada Hi Dharmasya Glaanir Bhavathi Bharatha |
Abhyudhanam Adharmasya Thadathmaanam Srijamyaham ||

Parithranaaya Sadhunaam Vinashaaya cha Dushkruthaam |
Dharma Sansthaapanaardhaaya Sambhavaami Yuge Yuge ||

To deliver the pios and to annihilate the miscreants, as well as to re-establish the principle of religion [Dharma], He will appear, millennium after millennium.
Where he clearly stated that he will descend himself to protect sadhus and to punish the one who perform Adharma, But In few extreme situations like in Kali Yuga where the one person has both the qualities of good and bad in him, in this case Lord accepted the Avatar as a Guru who can direct the person in the right way or to show the path of Dharma and remove Adharma.
For Example: Kapila Maharshi was the Kala Avatar of Lord Vishnu, Shankaracharya is the Kala Avatar of Lord Shiva.


When God empowers a living entity with some particular shakti /power, for some specific activity, and for wellbeing of everyone [Loka kalyanam] to represent Him, then the the person is called as Avesha Avathar in other words
When the Lord is present in someone by portions of His various potencies, the living entity representing the Lord is a Shaktyavesha Avatar, Lord Parashuram is the example of this Shakthyavesha Avatar.

[Also Read : 'Lord Rama' Defined By A Demon ]


In all the Avataras like Matsya, Kurma, Dhanvantari, Mohini etc., where the lord accepted the Avataras for particular purpose and again merge within Lord Vishnu which no longer be visible after the purpose is completed, but Purna Avatar is an avatar where the Lord is complete [with all the kalas], and not for one particular purpose.
For example, Lord Sri Ram is the Purna Avatar of Lord Vishnu, where he take birth as a human and even after the death of Ravana, he lived for 11,000 years as a king 
which is not only for the purpose. Therefore Lord Rama is worshipped as the Purna purush.


Example of Paripurna Avatar is Lord Sri Krishna, In the Case of Lord Rama he behaved completely as a normal human, but Lord Krishna never behaved so.
Lord Krishna has Ghora and Aghora Rupas [forms] and can be of many forms at a time.
For Examples; Sri Krishna showed different forms of Vishwarupa different times. He showed a different form of Vishwarupa to Devi Yashodha in his mouth, he showed different form of Vishwarupa in the Sabha of Dhrithrashtra, he showed a different form to Arjuna in Bhagavad Gita.
The Avtara of Lord Krishna can't be followed by humans. [one cannot do everything what Lord Krishna did as he didn't behaved as a normal human] 
while Purna Avathara like Lord Rama shows everyone the way of living life with Dharma and every individual should follow it.