Ganesh is one of the most respected Hindu deities. He is also one of the most easily recognised with his elephant head. He is the son of Shiva "the destroyer", one of the Hindu 'Trinity', and Parvati, his consort.
There are several traditions as to how Ganesh's elephant head came into being. One of the traditions tells of Shiva inadvertently killing a doughboy who had blocked his path into his own home. In remorse at having killed the doughboy, he took up the body but was unable to find the head, and so replaced it with a convenient head of an elephant that he had lopped off a few minutes earlier.
Another tradition tells of Shiva returning home after many years away on a hunting trip, and finding his (now grown-up) son in bed with Parvati. In a rage, believing that Parvati had taken a lover, he cut Ganesh's head off, and had to replace it with the head of the first animal that he saw - which was an elephant. A variant on this story suggests that Parvati requested Ganesh to guard her bathroom while she washed herself - and when Shiva returned, Ganesh did not recognise him, and again in anger Shiva cut off his head.
Ganesh is a very popular God, especially in his home city of Mumbai. He is the God of luck and fortune, of doorways, of the household, and of writing. He is also the remover of obstacles, and as such it is normal to worship Ganesh before any other Hindu God.
Ganesh is also known by other names in South India - such as Vinayaka and Aiyyapa.