The Forms and Incarnations of Ganesha

1. Bala Ganapati - "the Childlike"
2. Taruna Ganapati - "the Youthful"
3. Bhakti Ganapati - "Dear to Devotees"
4. Vira Ganapati - "Valiant Warrior"
5. Shakti Ganapati - "the Powerful"
6. Dvija Ganapati - "the Twice-born"
7. Siddhi Ganapati - "the Accomplished"
8. Ucchhishta Ganapati - "Lord of Blessed Offerings"
9. Vighna Ganapati - "Lord of Obstacles"
10. Kshipra Ganapati - "Quick-Acting"
11. Heramba Ganapati - "Protector of the Weak"
12. Lakshmi Ganapati - "Giver of Success"
13. Maha Ganapati - "the Great"
14. Vijaya Ganapati - "the Victorious"
15. Nritya Ganapati - " the Dancer"
16. Urdhva Ganapati - "the Elevated"
17. Ekakshara Ganapati - "Single-Syllable"
18. Varada Ganapati - "the Boon-Giver"
19. Tryakshara Ganapati - "the Lord of Three Letters"
20. Kshipra Prasada Ganapati - "the Quick Rewarder"
21. Haridra Ganapati - "the Golden One"
22. Ekadanta Ganapati - "Single Tusk"
23. Srishti Ganapati - "Lord of Happy Manifestation"
24. Uddanda Ganapati - "Enforcer of Dharma"
25. Rinamochana Ganapati - "Humanity's Liberator"
26. Dhundhi Ganapati - "the Sought After"
27. Dvimukha Ganapati - "Two Faced"
28. Trimukha Ganapati - "Three-Faced"
29. Sinha Ganapati - "the Fearless"
30. Yoga Ganapati - "the Yogi"
31. Durga Ganapati - " the Invincible"
32. Sankatahara Ganapati - "the Dispeller of Sorrow"

View an image gallery and read a description of each of Ganesha's 32 traditional forms.

The 8 Incarnations of Ganesha as described in the Mud gala Purana

" Like the Ganesha Purana, the Mudgala Purana considers Ganesha to represent the ultimate reality of being. As such, Ganesha's manifestations are endless but eight of his incarnations (Sanskrit:अवतार; avatāra) are of most importance. The eight incarnations are introduced in Mud.P. 1.17.24-28. The text is organized into sections for each of these incarnations. These are not the same as the four incarnations of Ganesha that are described in the Ganesha Purana."

"The incarnation described in the Mudgala Purana took place in different cosmic ages. The Mudgala Purana uses these incarnations to express complex philosophical concepts associated with the progressive creation of the world. Each incarnation represents a stage of the absolute as it unfolds into creation. The list below provides a summary of the philosophical meaning of each incarnation within the framework of the Mudgala Purana: Along with the philosophy, typical Puranic themes of battles with demons provide much of the story line. The incarnations appear in the following order:" -- http://en.wikipedia.org/wiki/Mudgala_Purana

1) Vakratunda - "twisting trunk"

The first in the series, Vakratunda, represents the absolute as the aggregate of all bodies, an embodiment of the form of Brahman. The purpose of this incarnation is to overcome the demon Matsaryāsura (envy, jealousy). His mount (vāhana) is a lion.

2) Ekadanta - "single tusk"

Ekadanta represents the aggregate of all individual souls, an embodiment of the essential nature of Brahman. The purpose of this incarnation is to overcome the demon Madāsura (arrogance, conceit). His mount is a mouse.

3) Mahodara - "big belly"

Mahodarais is a synthesis of both Vakratuṇḍa and Ekadanta. It is the absolute as it enters into the creative process. It is an embodiment of the wisdom of Brahman. The purpose of this incarnation is to overcome the demon Mohāsura (delusion, confusion). His mount is a mouse.

4) Gajavaktra - "elephant face"

Gajavaktra is a counterpart to Mahodara. The purpose of this incarnation is to overcome the demon Lobhāsura (greed). His mount is a mouse.

5) Lambodara - "pendulous belly"

Lambodara is the first of four incarnations that correspond to the stage where the Purāṇic gods are created.
Lambodara corresponds to Śakti, the pure power of Brahman. The purpose of this incarnation is to overcome the demon Krodhāsura (anger). His mount is a mouse.

6) Vikata - "unusual form", "misshapen"

Vikata corresponds to Sūrya. He is an embodiment of the illuminating nature of Brahman. The purpose of this incarnation is to overcome the demon Kāmāsura (lust). His mount is a peacock.

7) Vighnaraja - "king of obstacles"

Vighnaraja corresponds to Viṣṇu. He is an embodiment of the preserving nature of Brahman.
The purpose of this incarnation is to overcome the demon Mamāsura (possessiveness). His mount is the celestial serpent Śeṣa.

8) Dhumravarna - "gray in color"

Dhumravarna corresponds to Śiva. He is an embodiment of the destructive nature of Brahman.
The purpose of this incarnation is to overcome the demon Abhimanāsura (pride, attachment). His mount is a horse.

(Phyllis Granoff, "Gaṇeśa as Metaphor")

The 4 Incarnations of Ganesha as described in the Ganesha Purana

1) Mahotkata Vinayaka

Mahotkata Vinayaka, who has ten arms and a red complexion. Different sources list his mount (vāhana) as either an elephant or lion. He was born to Kashyapa and Aditi in the Krita yuga. The name Kāśyapaḥ (descendant of Kaśyapa) for Ganesha refers to this incarnation.
This incarnation killed the demon brothers Narantaka and Devantaka, as well as the demon Dhumraksha.

2) Mayuresvara

Mayuresvara, who has six arms and a white complexion. His mount is a peacock. He was born to Shiva and Parvati in the Treta yuga.
He incarnates for the purpose of killing the demon Sindhu. At the end of this incarnation he gives his peacock mount to his younger brother Skanda, with whom the peacock mount is generally associated.

3) Gajanana

Gajanana, who has four arms and was born with a red complexion. He has a mouse as his mount. He is born to Shiva and Parvati in the Dvapara yuga.
He incarnates for the purpose of killing the demon Sindura, who was so-named due to his reddish-pink complexion. It is during this incarnation that Ganesha gives the discourse known as the Ganesha Gita to King Varenya.

4) Dhumraketu

Dhumraketu is gray in colour, like ash or smoke (dhūmra). He has either two or four arms. He has a blue horse as his mount. He will come to end the decline of the Kali yuga. During this incarnation he kills numerous demons.
There is a parallel between this incarnation of Ganesha and the tenth and final incarnation of Vishnu, who will also come at the end of the Kali yuga, riding upon the white horse Kalki.

(John A. Grimes. "Ganapati: Song of the Self")