The term avatar is usually associated with Vishnu. His avatars are divided into different categories. There may be more and other assignments than those mentioned here.
Purnavatara, Amshavatara, Purushavatara, Lilavatara, Gunavatara, Manvantaravatara, Yugavatara, Shaktyaveshavatara, Bhagavadaveshavatara.
Purnavatara – Purna means full. Purnavataras are Rama and Krishna.
Amshavatara – Amsha means part. Amshavataras are mystics and sages.
Purushavatara – Purusha is the cosmic primal man
Karanodakashayi – Vishnu lying on the causal ocean.
Vishnu as the cause of being.
Garbhodakashayi – Vishnu lying on the primeval ocean.
From Vishnu’s navel grows a lotus, in which Brahma, the creator, sits.
Kshirodakashayi – Vishnu lying on the milky ocean.
The the universe sustaining Vishnu.
Lilavatara – Lila is the divine play.
Dashavatara.
Gunavatara – The nature’s fundamental forces, Gunas.
Brahma is Rajas – Activity – Creation
Vishnu is Sattva – Balance – Sustainability (Purushavatara Kshirodakashayi)
Shiva is Tamas – Inactivity – Dissolution
Manvantaravatara – Avataras of the 14 Manvantaras
Yajna, Vibhu, Satyasena, Hari, Vaikuntha, Ajita, Vamana, Sarvabhauma, Rishabha, Vishvaksena, Dharmasetu, Sudhama, Yogeshvara, Brihadbhanu.
Yugavatara – Avataras of the 4 Yugas
The Dashavataras are commonly assigned to the four Yugas 4 – 3 – 2 – 1.
Satya Yuga – Fish, Turtle, Boar, Manlion.
Treta Yuga – Dwarf, Rama with the Ax, Rama
Dvapara Yuga – Krishna, Buddha
Kali Yuga – Kalki
The Mahabharata says that Kali Yuga began after Krishna’s death. Then Buddha would have to be assignet to Kali Yuga.
Shaktyaveshavatara – Empowered avatar indirectly
Vishnu empowers a being and transmits his power to it.
Four Kumaras – To spread spiritual knowledge.
Narada – To proclaim Bhakti, the love of God.
Brahma – To create.
Vyasa – To compile the Veda.
Bhagavadaveshavatara – Directly empowered avatar
Vishnu himself enters a being and acts through it.
Kapila – Founder of Sankhya Philosophy and composer of Kapila Gita.