In Hinduism, Lord Ganesh is primarily worshipped as the elephant-headed deity, known for his wisdom, intellect, and the ability to remove obstacles. However, there are different forms and manifestations of Ganesh that are worshipped in various traditions.

Lord Ganesh worship is often centered around his well-known and beloved form as the remover of obstacles, the lord of wisdom, and the patron of new beginnings.

Here are eight forms or aspects of Lord Ganesh that are commonly recognized:

1. Vakratund:
   • “Vakratund” means the one with a curved trunk. This is one of the most common forms of Ganesh, and he is often worshipped at the beginning of any auspicious event to remove obstacles.
   • In one story, it is said that Ganesh’s curved trunk, which earned him the name Vakratund, was a result of a battle with a demon named Matsya. The demon assumed the form of a fish and tried to escape the grasp of Ganesh, who caught him and wrapped him around his stomach, causing the demon’s body to bend.
2. Ekadanta:
   • “Ekadant” refers to the deity with a single tusk. This form of Ganesh is depicted with one broken tusk, symbolizing sacrifice and renunciation of worldly attachments.
   • Lord Ganesha in this form, he had four arms, one tooth, huge belly and his head resembled that of an elephant. This avatar killed Madasur, a demon who was harming the gods. The gods had approached Lord Vishnu for protection. He told them to propitiate Lord ‘Ekadanta’ and seek his protection. ‘Eka’ stands for ‘Maya’ the ‘embodied’ and ‘Danta’ for ‘Truth’. Thus Ekadanta symbolises ‘the Supreme Truth which wields the Maya’. The gods then prayed for Lord Ekadanta. He finally appeared and assured them that he would end their miseries.
3. Mahodar:
   • “Mahodar” means the one with a big belly. Ganesh’s large belly is symbolic of his all-encompassing nature, indicating that he can digest both the joys and sorrows of life.
   • In this form of Mahodar who vanquished Mohasur, the demon of delusion and confusion. This demon had conquered all the three worlds and was causing huge problems to everyone. It was the constant praying of the gods and kings that pleased Lord Ganesh and he said that he personally would destroy Mohasur. He took the form of Mahodr and met Mohasur. The later, instead of fighting with Mahodar, sought forgiveness for his sins. Lord Mahodar was pleased and commanded him to go back to ‘Pataal Lok’ and never return again.
4. Gajanan:
   • “Gajanan” means the elephant-faced one. This is a general term used to describe Ganesh’s primary form, where he is depicted with an elephant head.
   • There is a story about the demon Gajamukhasur who had the head of an elephant. This demon gained a boon that made him powerful, and he started causing havoc. Ganesh, in his form as Gajanan, is said to have defeated Gajamukhasur, symbolizing the triumph of wisdom and righteousness over demonic forces.
5. Lambodar:
   • “Lambodar” means the one with a potbelly. Ganesh’s large belly is not only a physical attribute but also represents the storage of knowledge and wisdom.
   • This is another form of Ganesh to subdue Krodhasur, the demon of anger. Krodhasur was born out of the anger that Lord Shiv developed for Lord Vishnu who took to the form of ‘Mohini’ to subdue demons. Krodhasur also managed to get the blessings of Lord Sun and became a very powerful king. Seeing no other option the gods prayed and requested Lord Ganesh to help them. Lord Ganesh obliged and took to the form of Lamobodar to subdue Krodhasur. It should be stated here that ‘Krodh’ means anger. The human life is greatly influenced by ‘Krodh’ and Lord Ganesh plays a huge role in shackling it.
6. Vigneshwar:
   • “Vigneshwar” means the Lord of obstacles. This form of Ganesh is specifically worshipped to seek his blessings for the removal of obstacles and challenges.
   • This is the most popular form of Lord Ganesh. Riding the ‘Sheshnag’, Vighnaraj into battle with Mamatasur, and overcame him. It must be stated here that Mamatasur was created born out of the laughter of Goddess Parvati Devi. Initially Goddess Parvati named the male form ‘Mama’ and asked him to worship Lord Ganesh. However a chance meeting with Sambara Asura took things for the worst. Sambara Ausra managed to allure Mama into the Asuric cult and initiated him into the demonic methods of worship. Hence his name became Mamatasur.
7. Dhumravarn:
   • “Dhumravarn” means the one with a smoke-colored complexion. This form of Ganesh is associated with the color of smoke and is worshipped for spiritual growth and knowledge.
   • Lord Ganesha took this form to defeat the demon Ahamkarasur, the demon of self-infatuation. Mythology says that it is the deity of Sun which created this demon. One day the Sun sneezed and out of it was born a demon. He went down to the world of Asuras and Sukracharya gave him the name ‘Aham’ on account of his birth from the ego of the Sun.
8. Balachandr:
   • “Balachandr” means the one with a crescent moon on the forehead. While the crescent moon is more commonly associated with Lord Shiva, in some depictions of Ganesh, particularly in regional traditions, he is shown with a crescent moon on his forehead.

It’s important to note that these forms are not necessarily distinct avatars in the same way that Lord Vishnu has avatars(Dashavatar).  They are more like aspects or manifestations of Ganesh, emphasizing different qualities and attributes associated with the deity. Different regions and traditions may have their own variations in the worship and depiction of Lord Ganesh. Ganesh is primarily worshipped in his elephant-headed form, representing wisdom, intellect, and the removal of obstacles.