This festival is celebrated on Bhadarva Sud 4th

Ganesh's Birth
Once Lord Shiva left his usual dwelling on Mt.Kailas in the Himalayas, to meditate in one of the caves in the same mountain. Two friends of Parvati, his consort, suggested that she should have a 'gana' - attendant of their own, since they were not too happy with Shivaji's 'ganas'. Parvati agreed. Soon after, using her divine powers, she created a son, naming him Vinayak. She instructed him to guard their home.
When Shivaji returned Vinayak prevented him from entering. After some bickering, Shivaji beheaded Vinayak with his 'trishul' (trident). When Parvati heard of this she lamented intensely. Shivaji realised his error. To set things right he sent his men to the forest to fetch the head of the first living thing they met. They encountered a baby elephant and returned with its head. With his divine power, Shivaji placed the head over his son's body. Instantly Vinayak sprung to life. Henceforth Vinayak also earned the name of Gajaanan. 'Gaja' means elephant. Shivaji blessed him with a boon that people would first worship and offer him dedication in all their auspicious activities. Those who didn't would not attain success.
Bhadarva Sud 4th is the day Parvati was offering the final pujan to the dead Ganeshji, when he resurrected.

Ganapati Vivaha (Marriage)
When Shiva and Parvati decided to have their sons, Kartikeya and Ganesh married, they stipulated that he who circumambulated the earth first would be considered the best deity and would be married first. Kartikeya flew off on his vehicle - a peacock. Poor Ganapati's vehicle was a mouse which was no match for a peacock. Being the choicest devotee of God, Parvatiji consoling him showed Ganeshji a simpler and quicker way. She advised him to cicumambulate the cow since she also symbolically represents another earth and he had reap the same benefit as going round the earth. Another reference cites him circumambulating his parents.Consoling him, Parvati showed a short cut. She divulged that it was ordained in the scriptures that he who offered pujan to his parents and then circumambulated them received the same merit as he who went around the earth. Ganapati hence married first showing that one who obeys the wish of the choicest devotee of God or his parents attains his desired wishes. Hence Ganeshji is invoked first in all auspicious events, rites and rituals such as marriage, opening ceremonies, ground-breaking ceremonies, yagnas and so on.
The Shiva Puran cites another story. Prajapati had two daughters, Siddhi (wealth) and Buddhi (intellect). He approached Parvati and Shivaji for the girls' marriage to Kartikeya and Ganeshji. However both girls wished to marry only the latter. Thus they married him. Siddhi gave birth to a son named 'Shubh' (auspiciousness) and Buddhi to 'Labh' (merit).
Therefore when businessmen and merchants offer pujan to Ganeshji and Lakshmiji they write 'Shubh' and 'Labh' inside their account ledgers to invoke the two deities.

Sentiments
The first person to observe the 'vrat' (vow) of Ganesh Chaturthi was Chandra - the moon. After Ganeshji's fame as leader ('pati') of Shiva's 'ganas', hence Ganapati, he was travelling through the heavens. As he passed Chandra - who prided on his attractive features, he slighted Ganeshji's peculiar form. In return, Ganeshji cursed him, "You shall bear the fruit of your karma. Whenever somebody does your darshan he will be cursed too." Chandra begged for forgiveness. Ganeshji then advised him to observe the Ganesh Chaturthi vrat and was freed from the curse.
By being aware of the detrimental effects of false pride one should cultivate humility. The Skanda Purana mentions this sentiment of the festival.
The Vayu Puran advocates the observance of this festival by listening to the following relevant episode of Shri Krishna, to be relieved from false accusation:
When Shri Krishna was falsely accused of pilfering the Syamantak Mani (gem) he observed Ganesh Chaturthi and was freed from the false charge.
This festival also inspires devotees to inculcate two virtues; obeying the commands of God and His choicest devotee, and consolidating faith in them, just as Ganeshji had faith and trust in Parvati regarding circumambulating his parents.

Names of Ganeshji
Ekadant (one-toothed)

• To scribe the Mahabharat Katha Ganeshji removed one of his tusks to carve a quill from it. He then scribed the epic on palm leaves as the sage Ved Vyas recited it. The scribing took three years!
  
• In south India there is a belief that when ploughing first began on earth, it was Lord Ganeshji who first ploughed using one of his tusks.
  
• In the Uttar Ramayan, Brahmand Puran and Padma Puran there is a reference of a battle between Ganeshji and Parshuram. During the duel, the latter's axe fractured one of Ganeshji's tusks According to the Brahmand Puran, it was his left tusk.
  

• Lambodar - from the long - 'lambo', tummy - 'udar'.
• Gajkarna - from the large elephant 'Gaja' ears - 'karna.'
• Kapil - from his ruddy complexion.
• Vikat - heavy-bodied.
• Bhalchandra - having a Chandra (moon) on his forehead - 'bhal.'
• Dwaimatur - one who has 'dwi' - two, 'mata' - mother; Parvati - who gave birth to him and Malini - a demoness who nurtured him.
• Vakratund - one who breaks the ego of he who behaves anti-socially ('Vakra').
• Mudgal - In south India, a special 'Mudgal Puran' extols Ganapati's glory. It cites 32 names, while 'Shardatilak' lists 51 names.
• Vigneshwar - One who removes mayic obstacles - 'vignas'