Ganesh is embodiment of wisdom

Ganesha is the embodiment of wisdom. When you use your intelligence and intuition to make the correct decisions then Ganesha is fully operative. "na deva da??am-ādāya rak?anti pa?u-pālavat | ya? tu rak?itu? icchanti buddhyā sa?vibhajanti tam ||" Meanding: The gods do not protect people by taking up clubs like herdsmen; unto those however, whom they wish to protect they grant intelligence (to make the right decisions.). Here’s a good prayer to activate your intelligence and to sharpen your critical thinking:– "o? ekadantāya vidmahe | vakratu??āya dhīmahi | tanno danti pracodayāt ||" Meaning: May we know that one-tusked deity, let us contemplate the elephant-faced One; may that Great Tusker enlighten our intellects.

From a mythological and iconographical point of view Ganesha is the embodiment or personification of wisdom. From a ritualistic point of view he is the lord of obstacles - Vighnesha (due to pervasion of ignorance) and he is the remover of said obstacles through his quality of wisdom and is thus the first god to be worshiped in all auspicious ceremonies. From a philosophical point of view he is none other than the Supreme Brahman spoken of in the Vedas. The Ganapati Atharvashirsh Upanishad states this quite unambiguously:–

"tva? vā?gmaya?s-tva? cin-ma?ya? | tvam ānanda-ma?yas tva? brahma?-maya? | tva? sac-cid-ānandā 'dvi?tīyo?'si | tva? pra?tyak?a?? brahmā?si | tva? j?āna-mayo vij?āna?-mayo?'si || 4 ||"

Thou art speech, Thou art the Pure Consciousness, Thou art pure bliss, Thou art the Ultimate Reality (Brahman), Thou art the incomparable combination of being absolute, consciousness and bliss; Thou art the Supreme Being (Brahman); Thou art the sheath of pure Intelligence.

"sarva? jagad-idam tva?tto jā?yate | sarva? jagad-ida? tva?ttas-ti???hati | sarva? jagad-ida?tvayi laya?m-e?ya?ti | sarva? jagad-ida? tvayi? pratye?ti | tva? bhūmir-āpo'nalo'ni?lo na?bha? | tva? catvāri vā?k-padā?ni || 5 ||"

The entire universe was manifested from Thee, the entire universe exists in Thee. The entire universe will again dissolve into Thee, and return to Thee. Thou art the earth, water, fire, wind and ether. Thou art the four levels of sonic vibration.

"tva? brahmā tva? vi??us-tva? rudras-tvam indras-tvam agnis-tvam vāyus-tva? sūryas-tva? candramās-tva? brahma? bhūr-bhuva?? svarom || 6 "

Thou art Brahma (the Creator), Thou art Vishnu (the Preserver) , Thou art Rudra (the Transformer), Thou art Indra (Cosmic Mind), Thou art Agni (the Mystical Fire) and Thou art the Cosmic life force (Vayu) Thou art the Sun and the Moon, Thou art the Supreme Being, Thou art the three realms; the earth, the astral region and the heavenly region.

So the Ultimate Reality and Ground of all Being is the blessed and beautiful Ganesha – it is He alone that takes the form of Brahma, Vishnu and Shiva and in fact the entire universe.

Ganesha is inspired by cosmic phenomena, an event in the constellation of Krittika (Pleiades). And all cosmic phenomena are used in the traditions as a way of “thinking about” and understanding the nature of the ultimate reality, the unknowable infinity or Brahman. But the ancients are never satisfied to leave it at that. They used the smriti or narrative mnemonics to record the event. And as the event occurred in the vicinity of the constellation of Orion that is often taken to be a form of the cosmic function we call Shiva, the maintainer of the cycles, the event was recorded as a birth in the family of Shiva and Parvati. And since it was the second recorded event in the constellation of Krittika, it was recorded as the second child of Shiva and Parvati.

However, the objective was the same as the requirement that led to the creation of the abstract function Shiva, namely to give us another handle and a way to think about the nature of that ultimate reality. Ganesha is personified into numerous spiritual aspects and human ideals represented by the 108 symbolic names given to the deity form, the most popular of which are Ganapati, Kaveesha, Kripalu, Lambodhara, Mangalamurti, Mrituanjaya, Omkara, Siddhidhata, Siddhivinayaka, Vakratunda, Vignaharta, Vigneshwara,, etc.

Ganesha is considered as supreme Godhead by Ganapataya sect and some puranas/upanishads which are dedicated to him. As per one of the stories from these sects/books, Parvati prayed to the supreme lord Ganesha to be born as her son which he obliged. If you read Ganesh sahasranama/ashtotra, you will find a lot of goddesses mapped as his consort/siddhis. The Ganapataya sect was one of the major devotional sect (probably in Western India where he is still popular) - Adi Sankara created the panchayatana pooja (and later Shanmatha pooja) by merging this sect with other popular sects (Saiva, Vaishnava, Saktha, Saura and Kaumara/Shanmuka).

In the currently prevalent system, Ganesha represents Chit (wisdom) and hence is the cause & remover of obstacles. The story of his race with Skanda also symbolizes the importance of wit/wisdom in achieving quick results. His wives Buddhi and Siddhi/Riddhi also reflect his role as the lord of wisdom and obstacles, hence we pray to him first for right guidance (buddhi) and victory (siddhi) before worshipping other gods or starting any major activity.

Foremost, it is all Shiva's divine play. And for every deed Shiva performs there always exists an underlying welfare to living beings. Shiva fixed an Elephant's head to Ganesha. The Elephant was Gajasur, a great devotee of Shiva, who after performing years of penance, desired a boon of forever staying close to Shiva on Mount Kailash. Thus, Shiva gave moksha to the Gajasur, and the Elephant had such great virtue of offering his head to Lord Ganesh. Shiva is Pashupatinath, the God and saviour of all animals. So, Shiva granted moksha to Gajasur and even to the goat, by offering their heads to Lord Ganesh and Daksha Prajapathi respectively. Therefore there was no injustice done to the animals, instead they got Moksha, the ultimate bliss which every living being wishes for! Om Gan Ganapatye Namah