Avatar in Sanskrit means coming down.

However this does not convey the full import of the term.

This term denotes the manifestation of the Divine with Form.

Vedas,the authority of Hindus do not speak of Personal God’s.

Reality called Brahman is beyond Attributes.

However, considering the fact that human nature is limited and it is difficult to meditate on the abstract,the concept of Personal God came in.

It was first coined by Patanjali as Iswara under the category of Iswara Pranidhana.

However, Puranas speak of Avatars of the Godhood.

There are avatars of Gods.

Of the Trinity of Hinduism,Brahma ,Vishnu and Rudra/Shiva.

We have Avatars of Ganesha,Subrahmaya as well.

In the case of Vishnu and Brahma they are portrayed as being born of womb.

Shiva Avatars do not seem to have this attribute.

Shiva Avatars seem to occur in a flash and disappear as evidenced in the Shiva Avatars as described in the Thiruvilayadal Purana in Tamil.

This does not seem to be reflected in North Indian texts.

Some of Shiva Avatars mentioned in Thiruvilayadal Purana,are mentioned in Linga,Shiva Puranas.

In the case of Vishnu,there are different classification of the Avatars.

People generally speak of the ten Avatars of Vishnu.

Matsya, Korma,Varaha,

Narasimha,Vamana, Parashurama,

Rama,Krishna,Balarama and Kalki.

Of these the first five happened during Krutha)Satya Yuga,

Rama and Parashurama in Treta Yuga
Krishna and Balarama in Dwapara Yuga.

And Kalki is yet to manifest towards the end of the present time scale Kali Yuga.

Parashurama seems to be present Dwapara,and even in Kaliyuga as he is a Chiranjeevi, Immortal.

To understand this one has to understand the Hindu Time Scale.

Time, according to Hindus,is cyclic and not linear.

That is Time recoils on itself.

We have the same time scale appearing again and again.

For details please refer my articles on Time, Cyclic.

The time scales are,

Manvantara,

Kalpa and Yuga.

Based on this concept there are two two classifications of the Avatars of Lord Vishnu.

Manvantara Avatars and Kalpa Avatars.

Manvantara Avatars are,

Brahma’s day – 1,000 catur yugas – is divided into fourteen periods of the rule of Manus (fathers of mankind) known as manvantaras. The incarnations during these manvantaras are listed as follows:

Yajna
Vibhu
Satyasena
Hari
Vaikuntha
Ajita
Sarvabhauma
Rsabha
Visvaksena
Dharma
Sudhama
Yogesvara
Brhadbhanu.

2.Kapla/Yuga Avatars of Vishnu.

Four yugas ( Ages ) which come in cycles:
Satya yuga, lasting 1,728,000 years
Treta yuga, lasting 1,296,000 years
Dwapara yuga, lasting 864,000 years
Kali yuga, lasting 432,000 years

In each of these yugas the Lord incarnates with a different body color according to the yuga.

asan varnas trayo hy asya
grhnato ‘nuyugam tanuh
suklo raktas tatha pita
idanim krishnatam gatah

“Krishna appears as an incarnation in every millennium. In the past, He assumed three different colors – white, red and yellow – and now he has appeared in a blackish color.”
– Srimad Bhagavatam 10.8.13

In the Satya yuga a white avatara appeared to Kardama muni to establish meditation as the process for self-realization.

In the Treta yuga a red avatara appeared to Brahma to establish fire sacrifice as the process for self-realization.

In the Dvapara yuga a dark avatara (Krishna) appeared as the son of Devaki to establish temple worship as the process for self-realization.

In the Kali yuga a Yellow avatara appeared (Chaitanya Mahaprabhu) as the son of Sachi Mata to establish the chanting of the holy names (nama-sankirtana) as the process for self.

3.Purusha Avatars of Vihsnu.

Krishna first incarnates as the three purusha avataras namely:
Karanodakashayi Vishnu – “The Lord who is lying on the causal ocean”.
Garbhodakashayi Vishnu – “The Lord who is lying on the universal ocean”.
Kshirodakashayi Vishnu – “The Lord who is lying on the ocean of milk”.

In addition,Gaudiya Sampradaya of Vaishnavam lists three more.

4.Kumaras
Narada
Varaha
Matsya
Yajna
Nara-Narayana
Kardami
Dattatreya
Hayasirsa
Hamsa
Dhruvapriya or Prsniggarbha
Rsabha
Prthu
Nrsmha
Kurma
Dhanvantari
Mohini
Vamana
Bhargava Parasurama
Ramachandra
Vyasa
Prlambhari Balarama
Krishna
Buddha
Kalki

5.Guna Avatars based on Samkya concept of Disposition s,Gunas

Vishnu assumes this guna avatar post. Sattvik Guna = Mode of goodness = Maintanence
Brahma Rajik Guna = Mode of passion = Creation
Siva Tamas Guna = Mode of ignorence = Destructiion

6.

Shaktyavesha avataras: – Empowered Incarnations

There is no limit to the number of shaktyavesha incarnations. But some are mentioned in the vedic literatures as examples. They are of two kinds, direct and indirect.

When the Lord Himself expands displaying a particular power of His opulences He is known as Sakshat.

When He empowers a living entity with some particular sakti – power, for some specific activity, to represent Him , that living entity is called indirect or avesha incarnation.

Avesha incarnations:

Four Kumaras empowered with knowledge.
Narada empowered with devotional service.
Brahma empowered with creative power.
King Pritu empowered with power to maintain living beings.
Parasurama empowered with power to kill evil elements.
VedaVyasa to compile the vedas.

Source.

https://iskcondesiretree.com/page/six-categories-of-avatars

Of these Purusha Avatars explain Universal Time Scale so are the Kalpa Avatars.

You may note the Avatars under the categories overlap.

This is because of our classification.

I shall write on each classification and their philosophical implications.

In addition,we have Archavatara mentioned in Southern Tradition.

##My observations are in italics.