There are different stories behind ‘ekdanta Ganesh’, one of form of deity Ganesh !
Story # 1
According to the Padma Purana, once Lord Shiva’s ardent devotee, Parashuram went to meet Him in Kailash. Since, Lord Shiva was busy meditating, Lord Ganesha prevented Parashuram from entering the abode. Parshuram is known for his anger in Hindu mythology. So, when Lord Ganesha stopped him at the gate, Parashuram became furious and attacked the Lord with his axe. Seeing the axe, Lord Ganesha recognised it as his father’s axe since it was gifted to Parashuram by Lord Shiva so, Ganesh ji refrained from blocking the attack of the axe and allowed it to chop off one of his tusks. Thus, Ganesha came to be known as Ekdanta. Later Parashuram realised his fault and asked for forgiveness from Lord Shiva, Goddess Parvati and Lord Ganesha.
Story # 2
There is another story about his one tusk. A different legend narrates that when Ganesh Ji was asked to write down the epic of Mahabhaarat, dictated to him by its author, sage Vyaas, taking into note the enormity and significance of the task, he realized the inadequacy of any ordinary ‘pen’ to undertake the task. So he broke one of his own tusks and made a pen out of it. The lesson offered here is that no sacrifice is big enough in the pursuit of knowledge.
In the same reference, it is also said that as he was writing Mahaabhaarat, his pen broke, and he broke his one tusk to continue writing.
Story # 3
As per ancient legend, there was a demon called mooshikasoora, He was granted a boon that he can’t killed by a being which has taken birth. Ganesha was created by godess parvathi and life was poured into it in Shiva absence. After Shiva’s return to kailash, he was stopped by this boy at entrance. In anger, shiva beheaded the boys head and burnt it to ashes with his third eye. After hearing the truth from parvathi abt their son, he asked his followers to get the head of any livng being whose head is placed towards north direction. (This is the reason, hindu ppl don’t prefer sleeping with their heads in north direction.) The followers got the head of an elephant ( ghajsoora) and shiva got the head fixed to his son. As it is a elephant’s head he is called ganapathi or gajanna. Mooshikasoora’s evl were increasing day by day. Lord ganesh was sent for killing moshikasoora as he was the appropriate man for the same. Mooshikasoora’s wife was a gr8 follower of goddess parvathi. She had asked parvathi devi to protect her husband’s life. To kill mooshikasoora lord ganesh used one of his tooth as weapon. As parvathi devi was proteting mooshikasoora’s life his tooth could not kill him. Later mooshikasoora learnt from his mistake and asked Lord ganesha to make him his vahana. That is the mouse. As once the weapon release cannot be taken back, lord ganesha only had single tooth. hence he is called ekdant.
Story # 4
Indra, the god of the skies, invited Ganesha to a feast. Taking his vehicle (his mouse), Ganesha began his journey to Indra’s palace. Because the mouse could not carry all his weight, he lost his balance and fell. The moon, shining in the night sky, laughed at him. Angered by this insult, ganesha broke off his left tusk and threw it to the moon. And that’s why he may be called Ekadanta (means “with one tooth”). Ganesha also cursed the moon “that whoever looked at the moon on the ganesh chaturthi night, they would undergo privations.” Even today, devotees avoid looking at the moon on ganesha days.