The Eight Incarnations of Lord Ganesha are mentioned below:
The first incarnation is Vakratunda which means the One with the Curved Trunk. It is a personification of the form of Brahman. In the form of Vakratunda, Lord Ganesha defeated the demon of jealousy and envy, known as Matsaryasura. Thus, the Lord is considered as the destroyer of jealousy. In this incarnation, His vahana is a divine Lion.
Ekadanta is the second incarnation of the Lord which denotes that the Lord has a Single Tusk. He is a manifestation of the essential nature of Brahman. The Padma Purana states that once Lord Parashurama visited Lord Shiva in Mount Kailash. But Ganesha prevented him from entering. Thus Parashurama became infuriated and threw his divine axe at Ganesha, who knew it was granted by his father Shiva. Thus He refused to stop the sacred weapon and allowed the axe to chop one of his tusks. Hence he was named Ekadanta. Ganesha later used this tusk to script the epic Mahabharata. He acted as a scribe to Sage Ved Vyas. A mouse is his mount or vahana in this avatar. In this incarnation, Lord Ganesha fought against the demon Madasura, who was the embodiment of conceit and arrogance.
The Mahodara incarnation is an embodiment of the wisdom of Brahman. In this form the Lord is depicted with a pot belly and His mount is a mouse (shrew). Mahodara is an amalgamation of Vakratunda and Ekadanta forms. He is born in this avatar to annihilate Mohasura, the demon of confusion and delusion. Later the demon became a devotee of the Lord.
According to legends, Mohasura was also known as Daitya Raja or the King of the Asuras. He was a devotee of Surya Deva and dominated the three Lokas or worlds. All the sages, deities and gods were terrified of him. Then Lord Surya told the gods and sages to pray to Mahodara. Lord Ganesha was pleased with the worship and devotion of the sages and decided to eliminate Mohasura.
Lord Vishnu and Shukracharya advised Mohasura to surrender and pray to Mahodara. Eventually the demon surrendered to the Lord and praised him with utmost devotion. Mohasura asked for His forgiveness and promised to follow the path of righteousness. Lord Ganesha became pleased with his devotion and instructed him to return to Pataal Loka. All the sages and gods were relieved and praised Lord Mahodara.
The Gajanana or Gajavaktra incarnation is equivalent to the Mahodara avatar. Gajanana means the Lord with an elephant head. The combination of an elephant head with a human body is one of the unique physical features of Lord Ganesha. In this avatar, the Lord defeated the demon of greed, Lobhasura, who was the son of Lord Kuber. Gajanana mounts a mouse in this incarnation.
The Lambodara incarnation of Ganesha is considered to be similar to Shakti (Sakti), the pure power of Brahman. Lambodara, also known as Lambodar, refers to the lord as an elephant headed deity with a large stomach. The divine mouse, Krauncha, is portrayed as His vahana in this avatar. Lord Ganesha incarnated as Lambodara with the objective to eradicate the demon of anger known as Krodhasura.
In the sixth incarnation of Vikata, Lord Ganesha is considered to be a form of Surya. Vikata, meaning abnormal form, is a ferocious and dreadful deity. He possesses the body of a human with a head of an elephant. In this incarnation, a divine peacock serves as His vahana. Vikata is a manifestation of the illuminating nature of Brahmin. The Lord incarnated as Vikata to eradicate Kamasura, the demon of lust
Vighnaraja is the seventh avatar of Lord Ganesha and is a personification of the preserving nature of Brahmin. Ganesha is equivalent to Lord Vishnu in this form. Shesha or Shesha Naga, the divine serpent, is his mount in this incarnation. Lord Ganesha appears as Vighnaraja to defeat, the demon of ego and possessiveness, Mamasura.
The eighth and last incarnation of Lord Ganesha is Dhumravarna, which depict the Lord in a grey coloured form. It is an embodiment of the destructive nature of Brahmin. His mount is a horse in this incarnation. As Dhumravarna, Lord Ganesha is equivalent to Lord Shiva. Dhumravarna is born with the objective to defeat the demon of self-infatuation, pride and attachment known as Abhimanasura or Ahamkarasur.