Ganesha was created by his mother, Parvati to guard her while she was bathing. She did not want to be disturbed by her husband, Shiva. She instructed Ganesha not to let anyone in. When Shiva returned home and was denied entry to his wifes quarters, he immediately struck off Ganesha’s head. A terrible argument ensued between Shiva and Parvati, threatening to spiral completely out of control. Parvati could only be calmed down when Ganesha were given a new head. The head chosen was that of the first creature to appear—an elephant.
As for how he came to have a broken tusk...? In Nepal, the story goes that Ganesha insulted the moon for being too "full." The moon flew into a rage and struck off one of his tusks.
He is the god of new beginnings, the patron of arts,sciences and Travellers ...and a lover of sweets!