Ganapathi has the head of an elephant and has a mouse as a vehicle. The elephant is well known for its intelligence. It also acts as guide to mankind in the sense that it clears the way as it treads along in a dense forest. Similarly, in the forest of life, to clear the thorny bushes of pain, sorrow and suffering, one should tread along the path shown by intellect. Ganapathi has the elephant face as a symbol of high intelligence that can guide us on the right path. 

The mouse (mooshika) is a symbol of darkness. Because Vinaayaka has control over the darkness, He is described as the rider on a mouse (mooshika). The mouse is also a symbol of the sense of smell. It moves about following the direction of the smell (vaasana). Vinaayaka is the One who has mastery over our Vaasanas, that is desires and ignorance (represented by mooshika).

He is called Ganapathi because He is the supreme Lord of the Ganas (Ganas are the embodiments of the organs of perception - Jnanendriyas - and of action - Karmendriyas). "Ga" means Buddhi (intellect). "Na" means "Vijnana" (wisdom). "Pathi" means master. Hence, another meaning for Ganapathi is master of the intellect and wisdom. He is called Vinayaka because He has no master above Him. He is also known as Vighneswara as He is the remover of obstacles (Vighna). Lord Ganesha has all the five elements under His control. He is the embodiment of all potencies. He has no superior above Him. He is the Pranava Swarupa (embodiment of Aum). Hence, the first prayer is always addressed to Him.