Lord Ganesha has only one tusk and the second tusk is a broken one. There are two popular stories that explain why Lord Ganesh has a broken tusk. One story is associated with Lord Muruga or Kartik, Lord Ganesha’s brother. The other story is associated with the writing of the Mahabharata.
Legend has it that once Lord Kartik was writing Stri Purusha Lakshana – the attributes of a female and male. Lord Kartik purposefully did not invoke Ganesha before writing the treatise. Lord Brahma had specifically mentioned to Lord Kartik to begin only after invoking Lord Ganesh. And it is also a common practice in Hindu religion by beginning any new event after invoking Lord Ganesha.
Lord Shiva and Lord Vishnu soon appeared on the scene and stopped the fight between the brothers.
Lord Vishnu asked Kartik to give back the broken tusk to Ganesha. He agreed to do so on the condition that Ganapati should not let it fall from his hands. If he puts it down he will be burnt. This is the reason why Ganesha is seen holding the tusk.
Another story is associated with the writing of the Mahabharata. Sage Vyasa wanted someone to write down the Mahabharata while he was dictating it. Lord Ganesh agreed to write down the Mahabharata on the condition that it should be dictated without stopping. Sage Vyasa agreed and Ganesha began to write down the Mahabharat.
After some days of the writing the stylus used by Ganesh broke and as he could not stop writing he broke off one of his tusks and started writing with it. This symbolically explains that any amount of sacrifice is not too much to gain knowledge.