Shri Ganesha incarnated in 8 different Avatars at different times to destroy demons, each of which represents a different state of emotion.
Vakratunda Avatar was taken to kill Matsara, the demon of jealousy.
Vakratunda rides a lion & has a curved trunk.

Shri Ganesha incarnated as
Ekadanta Avatar to kill Mada,
the demon of vanity.
He has one tusk and rides a rat.
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Lambodara Avatar of Shri Ganesha was to kill Krodha, the demon of rage.
Lambodara has pendulous belly
his mount is a rat/ mouse.

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Dhumravarna Avatar of Shri Ganesha corresponds to Lord Shiva so is smoke-grey in colour
He incarnated to kill Ahamkara, the demon of arrogance.
His mount is a horse.

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Shri Ganesha incarnated as Mahodara Avatar to kill Moha,
the demon of attachment.
He has a big belly & His mount is a mouse.

Vighnaraja Avatar,
the King of obstacles came to kill Mama, the demon of self indulgence.
His mount is the celestial serpent Sesh Naag.
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Eephant head Gajanana Avatar who rides a mouse,
incarnated to Kill Lobha, the demon of greed.
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Vikata Avatar of Shri Ganesha is an unusual form, peacock is his mount.
He incarnated to overpower Kama, the lord of lust.
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