Four avatars of Ganesha are listed in the Ganesha Purana: Mohotkata, Mayūreśvara, Gajanana and Dhumraketu. Each avatar corresponds to a different yuga, has a different mount and different skin complexion, but all the avatars have a common purpose – to slay demons.
The Mudgala Purana describes eight avatars of Ganesha:
Vakratunda (Vakratuṇḍa) (“twisting trunk”), his mount is a lion.
Ekadanta (“single tusk”), his mount is a mouse.
Mahodara (“big belly”), his mount is a mouse.
Gajavaktra (or Gajānana) (“elephant face”), his mount is a mouse.
Lambodara (“pendulous belly”), his mount is a mouse.
Vikata (Vikaṭa) (“unusual form”, “misshapen”), his mount is a peacock.
Vighnaraja (Vighnarāja) (“king of obstacles”), his mount is the celestial serpent Śeṣa.
Dhumravarna (Dhūmravarṇa) (“grey color”) corresponds to Śiva, his mount is a horse.