There are many stories regarding his birth. The most popular one is as follows:
One day when Lord Shiva was away, Goddess Parvathi wanted to take bath in the river. To guard against any unwelcome intrusion, she took a little turmeric paste from her body and created a young boy. She gave him clear instructions not to allow any body inside. Unfortunately at this time Lord Shiva returned. The new boy refused him admission, though lord Shiva told him that he is the husband of Parvathi. Lord Shiva became very angry and cut off the head of this new boy and threw it away using his trident. At this time, Goddess Parvathi came back and became very sad, that the boy created by her was killed. Lord Shiva wanted to console Parvathi. So he sent his army to bring the first head that that they see on their way. The first head that they saw was the head of dying elephant. Shiva fixed that head on the boy’s trunk and brought him back to life. He also adopted him as his elder son and made him chief of his army.
There are many other stories on how Ganapathi was born. The one in Uthara Ramayana says that once Shiva and Parvathi took the form of elephant and were wandering in the forest. A son was born to them and that is Ganapathi.
Yet another story tells that Goddess Parvathi wanted a son badly and did Tapas to Lord Vishnu. He blessed her and a son was born to her. Goddess Parvathi invited all Gods to come and see her baby. When Lord Sani (Planet Saturn) came and had a look at the baby, the baby’s head was burnt. Lord Vishnu traveled throughout the world and brought back a head of an elephant which was fixed to the baby. All Gods present there blessed the baby, that unless he is worshipped first, they would not accept any worship.
Yet another story is that of Gajamukhasura. Gajamukhasura did great penance toward Lord Shiva. When he came before, he requested Lord Shiva to live in his belly. When Lord Shiva did not return, Goddess Parvathi sought the help of Lord Vishnu. Lord Vishnu dressed himself as a flutist and took along with him Nandi, the steed of Lord Shiva, Nandi was made to dance funny dances, in front of Gajamukhasura to the soulful music provided by Lord Vishnu (This type of bull dancing to the tune of music is still prevalent in South India). Gajamukhasura was pleased and gave a boon to Lord Vishnu, without knowing him who he is. Lord Vishnu requested him to release Lord Shiva. When Lord Shiva came out of his belly the asura died. While dying he requested Lord Shiva, that his head should be remembered after he is dead. After his death, Lord Shiva fixed the asura’s head on his elder son.
Another story tells that Shiva in a fit of rage killed Adithya, the son of sage Kashyapa. When Kashyapa became very angry, Lord Shiva replaced the head of Kashyapa’s son by the head of Indra’s elephant. The sage was infuriated and cursed that Lord Shiva’s son also will undergo the same fate as his own son.
Still another tale states that on one occasion, Parvati’s used bath-water was thrown into the Ganges, and this water was drunk by the elephant-headed Goddess Malini, who gave birth to a baby with four arms and five elephant heads. The river goddess Ganga claimed him as her son, but Shiva declared him to be Parvati’s son, reduced his five heads to one and enthroned him as the controller of obstacles (Vignesha).
Ganapathi lost his one tusk
The most popular story as to how Ganapathi lost his tusk is as follows. Veda Vyasa decided to compose the huge epic Mahabharata. He needed some body to write down his composition, as soon as he composes it. Ganapathi, being the God of knowledge, was requested to do this job. Ganapathi agreed on one condition. Veda Vyasa should dictate all he wants to write at one stretch and without break. Vyasa agreed but he put a condition that Lord Ganapathi should write a verse only after understanding it. After agreeing to this Ganapathi broke one of his tusks and used it as a pen. Veda Vyasa composed extremely difficult to understand verses, whenever he wanted time to think. This was the reason why Ganapathi become the one tusked God.
Padma Purana ascribes a completely different reason for this. It seems once Lord Parasurama went to Kailasa to see Lord Shiva. Lord Shiva was at that time asleep. Lord Ganapathi, who was guarding the room of Lord Shiva, did not allow Parasurama inside. A war ensued between Ganapathi and Parasurama and in this war Ganapathi lost one of his tusks because of the axe of Parasurama which had been gifted to him by Lord Shiva.
There is yet another story connecting moon to the loss of one tusk of Ganapathi. This is as follows. Once on a Vinayaka Chathurthi, Ganapathi got very large offering of very tempting Modhaks. He ate all that was offered and being unable to walk with a heavy belly, mounted on his steed, the mouse. Suddenly a snake appeared from the forest. The mouse on whom, Ganapathi was riding was terrified at the sight of a snake and started running away. Naturally Ganapathi fell from his steed and his belly broke slightly. Not loosing his presence of mind Ganapathi caught hold of the snake and tied his belly tightly. Chandra, who was seeing all this laughed at Ganapathi. The short tempered Ganapathi became very angry, broke one of his tusks and threw it at the moon. The Moon broke in to pieces. Ganapathi also cursed the moon, that, he would become a dark globe within 15 days, daily loosing one crescent and whosoever sees him on a chathurthi day (Fourth day of the waning moon) would be laughed at. Later Lord Shiva brokered a peace between Ganapathi and the moon. He said Moon will get reduced to dark as per Ganapathi’s curse and later regain his brilliance in another 15 days. (Some people ascribe the moon’s downfall due to a curse by Daksha, when he saw that out of his 27 daughters married to moon, he was showing partiality to only one girl Rohini).
Ganapathi made a speedier round of the world
Once sage Narada offered a very celestial mango fruit to Lord Shiva. Both his children Ganapathi and Subrahmanya wanted that fruit. Narada suggested that the one who travels all round the world first should get this fruit. Both Ganapathi and Subrahmanya agreed. Immediately Lord Subrahmanya started for his journey on his steed the peacock. Ganapathi who is very heavy and rides on a slow mouse circled round Lord Shiva and Parvathi and said that he has gone round the world because the entire world was within Lord Shiva and Goddess Parvathi. Lord Shiva was pleased with this explanation and gave the mango to Ganapathi
Another story says that there was a need for Shiva to appoint a commander and the above competition was held and Ganapathi won the post as explained in the last paragraph and was nominated as Ganesa or Ganapathi (Both meaning the Lord of Shiva’s army) and later Lord Subramanya was appointed as the captain of the army of devas.
Another story says that when Ganapathi and Subrahmanya reached marriageable age, Shiva held the above competition and Ganapathi won it as described and his marriage with Sidhi and Budhi was celebrated first.
Ganapathi blessed the South India with a great river
The marriage of Shiva and Parvathi was to be celebrated on mount Kailasa. All the people all over the world started traveling towards mount Kailasa. Due to the heavy burden on the northern side, the world started tilting towards the north. To correct this Lord Shiva requested sage Agsthya to go and live in the South of India. Agasthya obeyed the orders of Lord Shiva and reached the south. The entire south India was extremely dry at that time. Lord Shiva had sent some sacred waters of Ganga along with Agasthya, which Sage Agasthya was preserving very carefully in his pot. One day when was asleep, Ganapathi took the form of a crow and toppled Agasthya’s pot. From the Ganga started spreading in to a river. This river was called kaveri – She who was spread by a crow.
Another story says that it was not Ganges which was in Agasthya’s pot but his wife Lopa Mudhra who was in love with South India.
Ganapathi prevented Ravana becoming powerful
Once Ravana by his musical skill pleased Lord Shiva, Blessing him Lord Shiva gave a Shiva Linga (Athma Linga) and asked Ravana to take it Sri Lanka and preserve it there. He told Ravana that if that Shiva Linga reaches Sri Lanka, no body can win over Ravana. But he also told that he should never keep that Shiva Linga down on earth at any other spot while on his way. When Ravana reached Gokarna (A place in Karnataka) he wanted very badly to answer natures call, Ganapathi appeared there as Brahmin boy and agreed to hold the Shiva Linga for some time. He also told Ravana that if he does not come back by the stipulated time, he would keep the Shiva Linga there and go away. Lord Ganapathi delayed the coming back of Ravana and kept the Shiva Linga there and vanished. That place is called Gokarna. This effectively prevented Ravana to be ever victorious.
Ganapathi helped his brother marry Valli, a tribal princess
Subramanya first married the Devasena the daughter of Indra. Subrahmanya Later during his travels in south fell in love with Valli who was a tribal princess. Though Valli loved Lord Subrahmanya she was scared of him, her being a simple tribal girl and he being the celestial God. One day while Valli was walking in the forest, Lord Ganapathi took the form of an elephant and chased Valli. Subrahmanya in the form of an old man gave her protection and requested the elephant to go away. Then Lord Subrahmanya showed his real form and both of them got married.
Ganapathi and Kubhera
Kubhera was the God of wealth and was very proud of his wealth. He had slight contempt towards Shiva for living in such poor conditions on the top of the mountain. One day he invited Lord Shiva for a feast in his capital Alakapuri and also told that he has arranged for so much food that no body can eat it completely Lord Shiva said that he was very busy and sent his son Ganapathi instead. Ganapathi not only ate all the food that was prepared but also everything in sight in Kubhera’s town. Kubhera understood his folly and requested for forgiveness