Six Categories Of Avatars

1) Purusha avataras 
2) Lila avataras 
3) Guna avataras 
4) Manvantara avataras 
5) Yuga avataras 
6) Shaktyavesha avataras

1) Purusha avataras:

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".

2) Lila avataras:

A list of lila avataras (pastime incarnations) mentioned in the Srimad Bhagavatam:

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

3) Guna avataras:

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 = Destruction

4) Manvantara avataras:

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

5) Yuga avatrars:

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-realization. 
 

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.