Ganesha
Ganesha, the child of Siva and Parvati, is one of the most popular gods of India. He is commonly depicted as having the head of an elephant, a pot-bellied body (which in modern iconography is that of a child), a single tusk, and four arms - two displaying the noose and goad (ankusha) - two giving the gestures of allaying fears (abhaya) and granting boons (vara). He is sometimes shown wearing a serpent as a sacred thread and sitting on his vehicle, a mouse.
Attributes
The different features of Ganesha iconography can be interpreted in a variety of ways. Ganesha's elephant head for example, represents the qualities associated with elephants: auspiciousness, strength, wisdom and loyalty. Ganesha's elephant ears are said to symbolise the quality of discrimination - in the Ganesha Upanishad they are likened to "winnowing baskets" - sifting truth from non-truth.
There are a number of explanations for Ganesha's single tusk, one of which is that he broke off one of his tusks to use as a pen whilst transcribing the epic Mahabharata from the words of the sage, Vyasa.
There are a wide variety of myths concerning the birth of Ganesha. One story from the Shiva Purana is that Parvati originally created a 'doorkeeper' to guard her whilst she bathed. Siva arrived home, and wanted to visit his wife in her bath, but the 'doorkeeper' did not recognise him and so would not allow him to pass. Siva flew into a rage and started to attack the doorkeeper, and summoned his Ganas to assist him. Siva eventually defeated the doorkeeper by beheading him. Parvati heard the ruckus and was extremely displeased by Siva's actions. Full of remorse, the god commanded his Ganas to go forth and bring back the head of the first being they found - and the Ganas returned with the head of an elephant. Siva joined the elephant head to the doorkeeper's body and revived him, naming him Ganapati, making him the leader of his Ganas and declaring that all would have to worship Ganesha before beginning any undertaking.
The Tantric Ganesha
Scholars have identified at least three tantric lineages of Ganapatyas who worshipped Ganesha as the supreme deity, although information on them is scanty. Unlike other deities at the centre of Tantric practices, such as Siva or Lalita, Ganesha is not primarily concerned with transcendental 'illumination', but is a god of everyday concerns - good fortune, overcoming obstacles, and success in worldly endeavours.