Ganesha is called Ek-Daantay , the one with only one tusk.

1. GajjMukhaSur

Gajjmukha-Sur was a Detya (Demon) who was booned to be killed neither by Astra Nor Shastra (tool or Weapon). Priam-Mada the wife of Gajjmukha-Sur , was a devotee of Goddess Guri (Parvati) and she Keep Guri Vrat for the protection of her husband. Gajjmukha fought with Devtas (gods) , but none of them could kill him , because of the boon. Finally Ganesha, the son of Shiva and Parvati , fought with him and Ganesha Broke his tusk ,which was neither Astra nor Shastra, and shot it towards Gajjmukh-Sur. Goddess Guri through her Bengal on the flying tusk in air and Stopped it . Ganesha asked Her Mother the reason , She replied that how could she the prayers and Vrats of Priam-Mada. While the mother and son was Arguing , Gajjmukha-Sur thought its the right time to escape, He distinguished himself as a Mouse and tried to escape. Ganesha saw him and he kept his feet over Gajjmukha to stop him. As the feet of Ganesha touched the mouse( Gajjmukha) , the mouse attained Satt-Bhuddhi ( Good thoughts in mind). The mouse started Chanting names of Ganesha and beg forgiveness from Ganesha. Priam-Mada came to battel field to see her husband , she was shocked to see her husband as a mouse under the feet of Ganesha , She asked Goddess Guri for explanation. Guri explained that your husband is the Luckiest , non in the world has got the opportunity but your husband to be the Mount of Ganesha and stay with him Forever ,the world will pray him with Ganesha.

2. Fighting with Parshuram

Once Parshurama went to see Lord Shiva on Kailasha . Lord Ganesha came into his way, who was ordered by his mother Parvati to Gaurd . Parshurama threatned Ganesha and ask him to fight with him if he want to stop him from meeting Shiva. In this fight Parshurama cutt off Ganesha Tusk. when Goddess Parvati saw this she lost her temper. Goddess took the form of Adhhi Shakti in anger that even shiva was unable to please her. She told that Parshurama is not satisfied with the blood of Kastriya’s that he wants to harm my son. Ganesha asked his Mother to forgive Parshurama and he some how succeded in pleasing his mother. Parshurama was so pleased with Ganesha that he gave him his Parshu and blessed him.

3. Writing Mahabharat

Ganesha and Ved Vyas narrating epic

On the advice of Brahma ,Rishi Ved Vyas  went to Lord Ganesha and requested him to become his Scribe in writing the great epic Mahabharat. Lord Ganesha agreed but no kalam (peacock feather , or gold feather used for writing) could write for so long ,to complete the whole Mahabharat. Ganesha broke his tusk for writing the epic.