1. Vedic perspective:

The chief of gaṇa-s is known as Gaṇapati. Gaṇa not only refers to an attendant on Śiva, but also means troop, multitude, sum total of animate and inanimate beings. Pati means chief and therefore Gaṇapati literally means chief of both animate and inanimate beings. Hence, Gaṇapati is also known as Brahmaṇaspati in Vedic literature. The common verse that is known to us is this one.

गणानां त्वा गणपतिं हवामहे
कविं कवीनामुपमश्रवस्तमम्।
ज्येष्ठराजं ब्रह्मणां ब्रह्मणस्पत
आ नः शृण्वन्त्रूतिभिः सीद सादनम्॥

gaṇānāṁ tvā gaṇapatiṁ havāmahe
kaviṁ kavīnāmupamaśravastamam |
jyeṣṭharājaṁ brahmaṇāṁ brahmaṇaspata
ā naḥ śṛṇvantrūtibhiḥ sīda sādanam ||

Meaning: We invoke you Gaṇapati, embodiment of immeasurable creativity, wealth, and brilliance. Please listen to our prayers seeking your blessings and reassurance. Please come here and take your seat.

Is Brahmaṇaspati the same as Indra? No, say Vedas. Brahmaṇaspati is the lord of all the beings, including gods and goddesses and Indra only presides over winds. In some Vedic verses, Indra is also described as ‘mahāhastī’ (long hands), probably referring to the tusk. But, why Gaṇapati is first invoked in all rituals? There is a connection between Agni (god of fire) and Gaṇapati. Rig Veda (I.31.2) says, “O! Adorable God (referring Agni) you are the first and foremost essence of life; you enforce the eternal laws; you are the offspring of two mothers. Hence intelligent commence their work by worshiping you.” Agni is said to have two mothers, which refer to two araṇi sticks (used to kindle fire). The two mothers in fact represent heaven and earth or spiritual world and material world. Similarly, Gaṇapati also has two mothers, Durgā and Camuṇḍā. It is said that Agni is in the belly of Gaṇapati and that is why his belly too big. When Agni, who is the carrier of oblations to the respective gods and goddesses, is placed in the belly of Gaṇapati obviously he needs to be worshiped first. Otherwise, benefits of oblations will not be carried to the respective gods.

Apart from this, Gaṇapati is also known as Vigneśvara (vigna means obstacles) which means he is the lord of all obstacles. If he is not worshiped first there could be obstacles in rituals. Therefore, he is first invoked to remove all the obstacles. When he is pleased, he not only does not cause obstacles, but also ensures that there are no obstacles while performing rituals of any kind.

गजाननं भूतगणादि सेवितं
खपित्थ जम्बूफलसार भक्षितम्।
उमासुतं शोक विनाशकारणं
नमामि विघ्नेश्वर पादपङ्कजम्॥

gajānanaṁ bhūtagaṇādi sevitaṁ
khapittha jambūphalasāra bhakṣitam |
umāsutaṁ śoka vināśakāraṇaṁ
namāmi vighneśvara pādapaṅkajam ||

O! Son of Umā! the destroyer of sorrows, I prostrate at your lotus feet and be kind enough to remove all obstacles.

Several instances can be quoted from Vedas alone to prove that worshipping Gaṇapati is important to remove all types of obstacles.

2. Tāntrik perspective:

In Tantra also Gaṇapati is invoked first. But tantra attributes different types of Vināyaka-s and important among them are seven types of Gaṇapati-s, each representing sapta mātṛ-s. The seven Gaṇapati-s are Mahāgaṇapati, Ūrdhva, Piṅgala, Ucchiṣṭa, Lakṣmī, Haridrā and Vallabha Gaṇapati-s. These seven Gaṇapati-s are associated with sapta mātṛ-s - Brahmāṇī, Māheśvarī, Kaumārī, Vaiśṇavī, Vārāhi, Indrāṇī and Cāmuṇdā. These seven goddesses test our spiritual progress by causing impediments and obstacles. A man generally fails to stay connected with God when he is in distress. These sapta mātṛ-s test us from different perspectives. They cause pride, anger, illusion, greed, envy, carping and denigration. By causing these obstacles, they test our depth of devotion and hence they are also known as vigneśī-s (causing obstacles). Gaṇapati is said to be their guardian and protector. These sapta mātṛ-s along with Yogeśvarī (making a total of eight) cause supernatural powers (aṣṭamāsiddhi). These eight supernatural powers divided themselves into two major aspects of supernatural accomplishments and normal intellect. These two major portions later became two feminine deities known as Siddhi and Buddhi and became the consorts of Gaṇapati. Hence, worshipping Gaṇapati also gives supernatural powers, apart from removing all obstacles not only during rituals, but also during our entire life time.

3. Commonly known forms of Gaṇapati:

There are thirty two forms of Gaṇapati, which are generally known.

बाल, तरुन, भक्त, वीर, शक्ति, उच्छिष्ट, सिद्धि, द्विज, विग्न, क्षिप्र, हेरंब, लक्ष्मि, महा, विजय, नृत्य, ऊर्ध्व, वर, एकाक्षर,त्र्यक्षरक्षिप्प, परसाद, हरिद्रा, एकदन्त, सृष्टि, उद्दण्ड, सङ्कटहर, ढुण्ढि, द्विमुख, त्रिमुख, सिंह, यो,ग दुर्ग and ऋणगणपति.

Bāla Gaṇapati, Taruna, Bhakta, Vīra, Śakti, Ucchiṣṭa, Siddhi, Dvija, Vigna, Kṣipra, Heraṁba, Lakṣmi, Mahā, Vijaya, Nṛtya, Ūrdhva, Vara, Ekākṣara, Tryakṣara, Kṣippaparasāda, Haridrā, Ekakadanta, Sṛṣṭi, Uddaṇḍa, Saṅkaṭahara, Dhuṇḍhi, Dvimukha, Trimukha, Siṁha, Yoga Durga and Ṛṇa Gaṇapati.

4. Gaṇapati in Purāṇa-s:

Almost in every purāṇa, there is a reference to Gaṇapati. But there are two exclusive purāṇa-s for Gaṇapati - Gaṇeśa purāṇa and Mudgala purāṇa. These two purāṇa-s hold Gaṇapati as Brahman, the highest form of Divinity. That is why, praṇava (ॐ) is considered as the form of Gaṇapati. It is also said that Brahman wanted to take a particular form through which He would show that He is the root and true form of everything so that everyone can worship Him. The root of creation is praṇava. Of all the beings, only the trunk of the elephant if turned to right gives the form of praṇava.

There are several versions about Gaṇapati’s origin. The commonly known version is available in Varāha Purāṇa, which goes like this. Saints and gods once experienced various obstacles*.  So, they went to Śiva for solution. Śiva thought about the problem and found out that out of pañcabhūta-s ākāśa is not represented in physical form whereas the rest four have forms (air, fire, water and earth). He then remembered His promise to Brahmā that He would create ākāśa in the form ofGaṇapati. Then Śiva smiled and from His smile originated a very handsome and effulgent young man. Śiva’s Consort Pārvatī looked at the young man with bereft love. This made Śiva angry and made this handsome young man as a deformed person with elephant head. The basic principle in the creation of Gaṇapati is the mental union of Śiva and Śakti, in order to remove all obstacles in righteous deeds of all men. However there are several variations and interpretations. The point is that Gaṇapati is not normally born, but a mythological creation by Pārvatī and Parameśvara through Their mutual looks.

(*In general only those who are pious, religious, good in nature, god fearing, etc tend to suffer more; this is only to spend the remnants of karmic imprints; when his or her karmic imprints are about to be exhausted, one first transforms into a sthitaprajña and later evolves into a jīvanmukta and when all his karmic account is wiped out, he is liberated not to be born again. At the same time evil doers enjoy life with all luxuries and pleasures, as they continue to accrue karmas, which results in their repeated rebirths and associated pains.)

5. Gaṇapati and kuṇḍalinī:

Auvaiyār is a great saint and she composed a hymn called Vināyagar Agaval in Tamil. This hymn conveys many subtle things like kuṇḍalinī, liberation, etc.  Details can be read in the above link.

6. Gaṇapati’s vehicle:

Mouse is his vehicle or vāhana. There is a story about how Gaṇapati got mouse as his vehicle. Ganeśa was crying for a vehicle for him. He frequently disturbed his mother Pārvatī for this. Śiva on noticing that His Consort is being troubled by Ganeśa, an idea dawned on Him. There was a little mouse disturbing Śiva all the time. In whatever manner Śiva tried to send the mouse out, the mouse came back again. When Ganeśa cried for a vehicle, Śiva gave the mouse to him as his vehicle.

There is another story about mouse becoming his vehicle in Ganeśa Purāṇa. There was a celestial musician known as Krauñca. One day, in the court of Indra, Krauñca accidentally stepped on the foot of sage Vāmedeva, who became angry and cursed Krauñca to become a mouse. However, Krauñca became a huge mouse and ended up damaging everything. Once, he ended up stepping on the hermitage of sage Parāśara, with whom Ganeśa was staying and destroyed the hermit. Ganeśa in order to teach Krauñca a lesson unleashed his pāśa (noose) on Krauñca and brought him to his feet.  Ganeśa then told him, "Krauñca...you have caused a lot of trouble and you deserve a severe punishment. But since you ask for my forgiveness, I will pardon you and use you as my vehicle". However, when Ganeśa mounted on Krauñca, he could not bear the weight of Ganeśa. Krauñca pleaded for Ganeśa to become light weight so that he could support him. Lord Ganeśa obliged and since then, has been using the mouse as his vehicle.

But mouse represents earth element and mūlādhāra chakra also represents earth element. Ganeśa presides over mūlādhāra and this subtle significance is expressly conveyed through mouse. Hence, Ganeśa is also known as mūṣika vāhana (mūṣika means mouse).

7. Ganeśa festivals:

i) Saṁkaṭahara Caturthi:

This is also known as saṁkaṣṭit caturthi, which is observed every month during fourth lunar day of kṛṣṇapakṣa or waning moon period. It is a religious vow (vṛt) and during this day, one should abstain from eating from sun rise to moon rise (normally this time period could be around 14 hours). After having darśan of moon, one can break his or her fast. This vow is observed for one year.

ii) Ganeśa Caturthi:

This is one of the major festivals and this is celebrated on the fourth lunar day ofśuklapakṣa or waxing period of the moon. This is the day when Śiva declared to the world that Ganeśa is more powerful than other gods. This is the day when Ganeśa is believed to descend to the earth and shower his blessings on those who worship him by offering modaka (containing grated coconut, jiggery and rice flour).

Kāñci Mahā Svāami, one of the greatest saints of recent days spoke about modakam. “When we offer different food varieties as oblations, concerened gods take only the essence of these food stuffs. Only human beings eat cooked food. Vigneśvara eats modakam which is cooked twice. Rice flour is first cooked to make the covering for modakam. It is cooked again after covering the pūrṇam (pūrṇam is the mixture of grated coconut and jiggery) with the flour. Sometimes, the modakam is also fried in ghee for greater ritual purity. When modakam is offered as oblations during homa, it is fried in ghee. When it is not for homa and is only offered as naivedya, it is offered only as cooked modakam. He is called modaka-hasta (hasta means hand) and he is the only form of god who keeps a sweet in his hand (trunk).

8. Upaniṣad-s associated with Gaṇapati:

There are few Upaniṣad-s about Gaṇapati.

i) Gaṇapati Sūktam

ii) Brahmaṇaspati Sūktam

iii) Gaṇapatyupaniṣad. This is also known as Gaṇpatayatharvaśīrṣam

iv) Gaṇeśapūrvatāpinyupaniṣad (pūrvatāpanī means first half)

v) Gaṇeśottaratāpinyupaniṣad

vi) Herambopaniṣad

9. Gaṇapati mantra-s:  

i) Mahāgaṇapati:

ॐ श्रीं ह्रीं क्लीं ग्लौं गं गणपतये वरवरद सर्वजनं मे वश्मानय स्वाहा ॥

om śrīṁ hrīṁ klīṁ glauṁ gaṁ gaṇapataye varavarada sarvajanaṁ me vaśmānaya svāhā ॥

ii) Śrī Vidyā Gaṇapati Vāñcha kalpalatā:

Please read in this link :

Śrī Vidyā Gaṇapati Vāñcha kalpalatā mantra