The great sage Ved Vyasa wished to compose an epic as vast as the Mahabharata, but he needed a scribe who could write it down. On the advice of Lord Brahma, he prayed to Ganesha.
Ganesha set a condition: "Once I begin writing, I will not stop for even a moment." Vyasa set a condition in return: "Write each verse only after you have understood its meaning." So from time to time Vyasa would recite a difficult verse, and while Ganesha pondered its meaning, Vyasa had time to compose new verses.
In the middle of writing, his pen broke. So that he would not have to stop, Ganesha broke off one of his own tusks, made it his pen and kept on writing. And so he came to be called "Ekadant", the one with a single tusk. (Some stories instead say his tusk was broken in a battle with Parashurama.)