The most fundamental of Hindu deities, is the Trinity of Brahma, Vishnu and Shiva - popular deities include Ganesha, Krishna, Hanuman and goddesses like Lakshmi, Durga,Saraswati.
There are several legends as to how Ganesha broke one of his tusks, giving him the name, Ekadanta, the one with a single tusk.
The first pertains to his battle with Parashurama. Parashurama was one of the incarnations of Vishnu, born on earth to teach a lesson to the ruling classes, the Kshatriyas, who had become very arrogant and were riding rough-shod over the ordinary people. As a human on earth, he meditated on Shiva and obtained the divine axe, Parashu, with whose help he waged wars against all the erring princes and cleared the world of their evil Deeply indebted to Shiva, he then came to Mount Kailas to pay obeisance to his mentor. Ganesh, who was guarding the entrance to his father's chambers, would not let him in, saying he had to wait till he obtained Shiva's permission. Parashurama felt that he, a devotee, needed no permission. When Ganesha refused to give in, Parashurama, hot-tempered at all times struck Ganesh's tusk with his axe and broke it. Shiva and Parvati appeared before him and chastised Parashurama, who then worshipped Ganesh and obtained his forgiveness and blessings .
According to another legend, Ganesh broke his tusk himself in his war with Gajamukhaasura. Gajamukha, a demon, did severe penance on the advice of Shukraachaarya, the guru of the asuras or demons, and obtained invincible powers from Shiva. He used these powers to harass the gods who then rushed to Ganesh for help. Ganesh battled with the demon but realised that, thanks to the powers given by Shiva, the evil one could not be killed Ganesha then broke his right tusk and threw it at Gajamukha cursing him to change into a mouse.He then got on to the back of the mouse and made it his vehicle, thereby keeping it under his control. According to another Puranic story, Ganesha's rat was really the Gandharva, Krauncha. Once, in the court of Indra, the king of the Devas, Krauncha insulted the sage, Vaamadeva. He was then cursed by the latter and turned into a large rat. The rat, true to its nature, entered the ashram of Sage Paraashara and caused great havoc to his dwelling as only a rat can. The Rishi then prayed to Vinayaka to save his simple dwelling. Ganesha appeared, made the rat his vehicle, and brought him under control As will be noticed, whatever the Puranic story, the purpose of making the rat his vehicle as to keep the rodent, whose nuisance value is high, under the control of Ganapati.