The eight incarnations of Ganesha

Vakratunda
1. Vakratunda ("twisting trunk"), first in the series, 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.
Ekadanta
2. Ekadanta ("single tusk") 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.
Mahodara
3. Mahodara ("big belly") is a synthesis of both Vakratunda 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.
Gajavaktra
4. Gajavaktra (or Gajānana) ("elephant face") is a counterpart to Mahodara. The purpose of this incarnation is to overcome the demon Lobhāsura (greed). His mount is a mouse.
Lambodara
5. Lambodara ("pendulous belly") is the first of four incarnations that correspond to the stage where the Purānic 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.
Vikata
6. Vikata (Vikata) ("unusual form", "misshapen") 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.
Vighnaraja
7. Vighnaraja (Vighnarāja) ("king of obstacles"), corresponds to Vishnu. 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 Śesha.
Dhumravarna
8. Dhumravarna (Dhūmravarna) ("grey color") 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.