In many idols of lord Ganesha, I saw that the left tusk is broken.
What is the story behind that broken tusk? Does it signify something?
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The story of how Ganesha broke his tusk is recounted in this excerpt from the Upodghata Pada of the Brahmanda Purana. Parashurama, the axe-wielding incarnation of Vishnu, had sucessfully defeated his enemy Kartavirya Arjuna and the kings allied with him, and so he wanted to thank Shiva for giving him the power to fight these enemies. Parashurama went to Mount Kailash to pay his obeisances to Shiva, but Ganesha stopped him, saying his father was sleeping along with his mother, and he didn't want Parashurama intruding on them in case they might be engaged in amorous pursuits. Parashurama was enraged that he was being prevented from seeing Shiva, and so he started fighting Ganesha. Ganesha was winning handily, but then Parashurama threw his axe at Ganesha and Ganesha didn't fight back against it, because the axe was a gift from Shiva:
Perceiving that the axe had been given to him by his father, Ganesha became desirous of meaning it not to go in vain. Hence he received it with his left tooth (tusk). Chopped off by the axe, the tusk fell on the ground, covered with blood like a mountain that fell on the ground when struck by Indra's thunderbolt.
Note that there are other stories 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. 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. So the most prominent story that's actually grounded in scripture is the one I gave above.
The broken tusk comes from Mahabharata. It goes as follows:
Maharshi Ved Vyas was instructed to write the epic by the Gods. He wanted it to be written down by the most knowledgeable one in the universe. Lord Brahma asked Maharshi to visit Lord Shiva and beg for his son SIDDHIDATA GANESHA to be allowed for the task. Lord Ganesha had a clause: While reciting the verses, if Ved Vyas stopped for a while, and Ganesha would have to stop because of the same; he would terminate his writing of the epic and Maharshi would have to search for another writer.That is, Maharshi would have to recite the entire epic at one go, without pausing at all.
Maharshi Ved Vyas agreed to this. However he himself put forth another condition: He told the Lord that he would have to understand every hymn, every verse before penning it down. He put this condition with the idea that he would be reciting something very tough; and while Ganesha would be pondering upon its meaning, he would get a scope to take a few moments of rest
However that was not the case with Lord Ganesha. He completed penning down the sacred hymns even before the sage had thought of the next. In the mean time, the pen He used for writing down the verses, began to wear away. Aware of His earlier condition, and not finding out any other alternative, the Lord pulled out his left tusk and used it to complete writng the great epic Mahabharata.
This is the story behind Lord Ganesha being EKADANTA.
It seems once Lord Parasurama went to Kailasa to see Lord Shiva. Lord Shiva was at that time asleep. Lord Ganapati, who was guarding the room of Lord Shiva, did not allow Parasurama inside. A war ensued between Ganapati and Parasurama and in this war Ganapati lost one of his tusks because of the axe of Parasurama which had been gifted to him by Lord Shiva.
It's also said that Ganesha allowed Parasurama to attack on him by Axe because he can't disrespect the Axe which was given by his own father.
The same source says that there are three stories for the broken tusk.One being same as @Shamayeta said. Third being a little variation of Ganesha's curse story. In this version, Ganesha before cursing moon throws his one tusk on him and breaking him in two pieces.
Skanda Purana narrates different story for his broken tusk. It says Lord Ganesha broke his own tusk for YOGADANDA.
He (Ganesha) pulled out his own huge tooth for sake of "Yogadanda" (the mystic wand of Yogic practice) and held it in his hand.