Lord Ganesha is the elephant-headed God.

The son of Parvati and Shiva.

He was created by Parvati to Guard her door, and was decapetated by Lord Shiva when he was tring to stop him from going in. when Parvati saw that her son was killed, she got furious, and to please her, Lord Shiva brought him back to life, and placed on his neck the head of the first living being that was found around- an elephant.

Lord Ganesha is called also Vigneshvara 'Lord of the obstacles'.  He is removing obstacles from the way of whom who worships him, and create obtacles on the way of thouse who needs to be checked. Therefore he is worshiped at every begining.