Ganesha means God of the masses or groups. 
His name is composed of GAN = masses/ groups and ESHA = God, meaning that He is the God of humanity and all living and mythical beings.
Ganesha symbolizes the union of Shiva and Shakti – the supreme Male and Female energies of cosmic creation – and being their Son is born with Shiva’s knowledge, and Shakti’s strength
The elephant head symbolizes many things per different texts, one of which is STRENGTH, to crush obstacles and barriers, therefore HE is also called “Vighna-Harta” which means Remover of Obstacles.
According to the 1400 AD Mudgala Purana, Ganesha’s 8 avatars (incarnations) are:
- Vakratunda (“twisting trunk”), overcomes the demon Matsaryāsura (envy, jealousy)
- Ekadanta (“single tusk”) conquers the demon Madāsura (arrogance, conceit)
- Mahodara (“big belly”) vanquishes the demon Mohāsura (delusion, confusion)
- Gajavaktra (“elephant face”) tames the demon Lobhāsura (greed)
- Lambodara (“pendulous belly”) controls the demon Krodhāsura (anger)
- Vikata (“unusual form”) destroys the demon Kāmāsura (lust)
- Vighnaraja (“king of obstacles”), eliminates the demon Mamāsura (possessiveness, attachment)
- Dhumravarna (“grey color”) subdues the demon Abhimanāsura (pride)
The Mugdala intrinsically links Him with Buddhi, Riddhi and Siddhi meaning He grants Wisdom, Affluence and Spiritual Power to those who subdue their baser self.
Adi Shankara’s Maha Ganesha Pancharatanam is a powerful prayer said to grant wishes, Siddhi Vinayak Mandir in Mumbai is one of His famous Temples, and one of His mantras is:
SarvaShiva 