There are various anecdotes which explain how Ganesha broke off one of his tusks. Devotees sometimes say that his single tusk indicates his ability to overcome all forms of dualism. In India, an elephant with one tusk is sometimes called a "Ganesh".
We all know about Ganesh Ji and his various stories like how he was born, how he got his elephant head and how he became the most popular Hindu Deity.
But do you know why our dear Lord Ganesha has a broken tusk?
So there are the two stories associated with the broken tusk of Shri Ganesh.
One day, as Ganesh was guarding his father who was in one of his deep meditations, a family friend came to visit.
It was Parashurama an incarnation of Lord Vishnu who is known for his serious temper.
Ganesh was a little perplexed because he was told to not let anyone disturb his father in meditation, however here was a serious VIP in the house.
Unable to disobey a direct order from his father he blocked Parashurama’s way, and as a result Parashurama lost his temper and threw an axe at Ganesh.
Ganesh knowing that the axe was a gift from his father allowed the axe to hit him out of respect.
The result: a broken tusk!
There is one more story about Ganesha losing his tusk, most famously the story included in some manuscripts of the Mahabharata concerning Ganesha breaking his own tusk off to continue writing the Mahabharata as Vyasa was dictating it to him.
When Vyasa wanted to start reciting, Ganesha had a banana leaf but no stylus. Not wanting to keep Vyasa waiting, he broke off his own tusk to use as a writing tool.
But this story is widely agreed by scholars to be a later interpolation, and it's not found in any critical editions of the Mahabharata, and you can see for yourself that it's not even included in this translation of chapter 1 of the Adi Parva.
All of these stories fall under the broad umbrella of Puranic literature, which are region and sect-specific collections of folk tales, essentially.
Their significance is generally more in their allegorical content than historical accuracy.