Ekadanta is another name for Gaṇeśa (Ganesa), son of Pārvatī (Parvati). Gaṇeśa is elephant-headed, and the left tusk of the elephant-face was broken’ Hence he came to be known as Gajānana. In several Purāṇa, there are different versions of legends, describing how this tusk got broken. The most circulated legend is as follows —
After slaying Kārtavīryārjuna (Kartaviryarjuna), Maharṣi (Maharshi; Great Sage) Paraśurāma(Parasurama) came to Kailāsa (Kailasa)to pay his respects to Mahādeva (Mahadeva). At the gate of the abode of of Mahādeva, Gaṇeśa stopped him. He asked Paraśurāma to wait, because Mahādeva and Parvatī, at that time, were at their private, inner quarters, and they should not be disturbed. But Paraśurāma was determined to visit Mahādeva at that very moment. This led to an argument and quarrel, which turned into a battle. Fighting against Gaṇeśa, Paraśurāma hurled at him the kuṭhāra (kuthara, axe), which was given to him by Mahādeva himself. If that weapon blessed with the divine power of Śiva failed, it would be a violation of Śiva’s grace– thinking thus, Gaṇeśa received the blow of the axe, upon his tusk. His left tusk was cut down at this blow. Since then, his name was Ekadanta.
However, in Śivapurāṇa (Sivapurana) it is said that Śiva (Siva), in his anger, behaded Gaṇeśa. Seeing his son slain, Pārvatī became furious in grief, and she was about to destroy the whole creation. The gods and Mahādeva himself, consoled her with great efforts and prayers. Then they were trying to find a way of rejuvenating Gaṇeśa. Under the instructions of Śiva, the gods went northward, and they found an elephant with one tusk. They beheaded the elephant and bringing that head, added it ti the headless torso of Gaṇeśa. Thus Gaṇeśa was restored to life, with the head of an one-tusked elephant, and so he became known as Ekadanta. [See Gaṇeśa]
In Vṛhaddharmapurāṇa (Vrihaddharmapurana) it is described that Nandī (Nandi), instructed by Śiva, was searching for a head for replacing it to Gaṇeśa’s body; and he found Airāvata (Airavata) on the way. When he tried to behead Airāvata, he had a fight withthe great elephant. As a result, one of the tusks of Airāvata got broken. Nandī brought that head of Airāvata and placed it upon Gaṇeśa’s headless body. That is why Gaṇeśa is known as Ekadanta.