Sage Mudgal has written the 'Ganeshsahastranam' containing the thousand Names of Lord Ganesh. The Dvadashanam verse (stotra) contains the following twelve Names of Ganapati.
प्रथमं वक्रतुण्डं च एकदन्तं द्वितीयकम् ।
Meaning: First Vakratunda, second Ekadanta
The meaning of these twelve names and some others is given below.
Generally it is considered that Vakratunda means one with a crooked mouth or trunk. However, this is incorrect. 'वक्रान् तुण्डयति इति वक्रतुण्ड: ।', meaning Vakratunda is one who punishes those who follow the wrong (unrighteous) path and leads them onto the righteous path. Vakratunda is one who straightens the tiryak and visphutit frequencies, that is the crooked raja-tama predominant 360 frequencies by means of His trunk and makes them sattvik (sattva predominant) like the 108 frequencies.
This Name is bestowed because He has only one unbroken tusk (the other is broken). Of the two tusks, the right tusk is complete while the one on the left is broken. The right side represents the surya (sun) channel (nadi). As the surya channel is effulgent, the tusk on this side of Lord Ganapati can never be broken. It is symbolic of the singular Brahman (God principle). The word dantin (दंतीन) is derived from the root dru-darshayati (दृ-दर्शयति) [meaning to show]. Thus the Name also means that He is the one who shows the direction to acquire the spiritual experience of Brahman which is non-dual. According to one school of thought, medha and shraddha (faith) are the two tusks. Medha means intellect, the ability to comprehend. Medha is the incomplete (broken) tusk and shraddha the complete one.
This word is derived from Krushna (कृष्ण) + pinga (पिंग) + aksha (अक्ष). Krushna means the one with a dark complexion, pinga means smoky and aksha means the eye. The dark complexion is in the context of earth while smoky refers to the clouds. Thus it means one who has the earth and the clouds as the eyes, that is one who can view everything on the earth and the clouds.
Gaja means cloud which is considered to be the representative of the dyu region (devlok) - the region of the deities. Vaktra means mouth. Thus Gajavaktra is one whose mouth is constituted by the dyu region (expansive). If Om (स्) is placed vertically, one gets the experience of Gajavadan (Ganapati). The Mudgal Puran has explained the meaning of the word 'gaja'as ga = the principle wherein everything undergoes dissolution and ja = the principle from which everything is created. So Gaja means Brahman (God principle).
Lambodar is derived from the words lamba (large) and udar (belly). Saint Eknath has explained the meaning of this word as,
The entire animate and inanimate creation dwells within You. Hence You are called Lambodar. - Shri Eknathi Bhagvat 1:3
According to the Ganapatitantra,Lord Shiva played the Damaru (a small hour glass shaped drum). Lord Ganesh grasped the knowledge of the Vedas through the deep sound of the Damaru. He learnt dancing by watching the TanDav dance everyday and music from the sound of the anklets of deity Parvati. Since He imbibed such varied knowledge, that is digested it, He developed a large belly.
Vi (वि) + krut (कृत) + akat (अकत) [akuti]. Vi means in a specific manner, krut means done and akat means the Final Liberation (Moksha). Hence, Vikat means the one who generates frequencies in a specific manner and bestows the Final Liberation.
Vighna (विघ्न) + ish (ईश) = Vighnesh (विघ्नेश). The word 'vighna' is derived from 'visheshen ghnati' which means mainly distress. The one who controls and destroys obstacles is Vighnesh. The obstacle in this context is entrapment by the 360 (raja-tama) and 108 (sattva) frequencies. This is contrary to the aim of traversing beyond the three components (trigunatit). Ish (ईश) is derived from i (ई) + sha (श). I-ikshate (ई-इक्षते) means to watch and sha-shamayate (श-शमयते) means to cool. So Ish is the one who keeps a watch and destroys the heat generated by the 360 and 108 frequencies. Vighnaharta (vanquisher of obstacles) is another Name of Ganapati. Since Ganapati gets rid of obstacles, He is worshipped before performing any auspicious religious ceremony.
Dhumra means smoke. Smoke is the initial state of materialisation. It is the transitory state between the solid manifest (sagun) and the unmanifest (nirgun) states. Thus, one who possesses such a smoky complexion is Dhumravarna. According to the principle that 'where there is smoke there is fire', Ganapati also possesses the fire element [embers (angar)].
Bhal means the forehead. The frequencies arising from Prajapati, Brahma, Shiva, Vishnu and Minakshi merge into one another and produce many groups of thousands of frequencies. Though Prajapati, Brahma, Shiva, Vishnu and Minakshi are unmanifest (nirgun), Their frequencies are composed of the three components (gun). The point of origin of three of these frequencies namely those of affection, mercifulness and motherly love (ahlad) is referred to as the moon (chandra). Thus, the one who adorns such a 'moon' on the forehead is Bhalchandra.
Actually this is the Name of Lord Shankar. But since Ganapati is His son, it also came to be linked with Him !
The word 'vinayak' is derived from 'visheshrupen nayakaha'. It means one who is endowed with all the features of a leader (nayak). 'It is universally accepted that there are six Vinayaks. The summary of information on Vinayaks given in the Manavgruhyasutra and Baudhayangruhyasutra is that the Vinayak attendants (gan) create obstacles and are troublesome and cruel. When they start harassing, people begin to behave as if insane, get horrifying nightmares and constantly harbour fear. To overcome this obstacle posed by the Vinayak attendants, the scriptures have advocated various rituals (Shantividhi). Vinayak, that is Ganapati is the presiding deity of these Vinayaks (who bring about destruction).'
'According to the information given in the Manavgruhyasutra, there are four Vinayaks - 1. Shalakankat, 2. KushmanDarajputra, 3. Urimat and 4. Devjayan. Sage Yadnyavalkya has mentioned sixVinayaks - 1. Mit, 2. Sammit, 3. Shal, 4. Kantak, 5. KushmanDa and 6. Rajputra. Lord Rudra and Lord Brahma appointed the Vinayaks as the presiding deities of the attendants and entrusted them with the task of posing obstacles in various human activities. Although Vinayak is ferocious by nature yet He bestows happiness when appeased. According to the Shaiva sect, Ganapati is the mild form while Vinayak is the ferocious form. The Smrutis Purans (mythological texts), stone engravings, etc. mention that Ganapati and Vinayak are one and the same; but it cannot be said when this concept became prevalent.' (Reference - Abhinav Marathi Dnyankosh, part four.)
Refer point '1. Origin and meaning'.
Gaja means elephant and anan means countenance. Thus Gajanan is one whose countenance is like that of an elephant (and whose body constitutes the entire universe).
'In the Ganapatyatharvashirsha,Ganapati is offered salutation as "Namo Vratapataye". Vratapati is the chief of the vratyas. The Aryans bestowed the name vrat, on the group of non-Aryans who favoured vowed religious observances (vrat) and did not believe in sacrificial fires (yadnya). The people belonging to this group are vrat. "व्रात इव व्रात्यः ।", meaning those like the vrat are vratya. A twice born (dvij) who does not have any sanskars (rites) performed on him is called a vratya.'
Chintamani is another Name of Lord Ganapati. An extroverted tendency (kshipta), a more extroverted tendency which may even lead to sin (mudha), an introverted tendency (vikshipta), a more introverted tendency (ekagra) and a state in which all activities have ceased (niruddha) are the five states of the subconscious mind (chitta). [Information on this is provided in 'Science of Spirituality : Vol. 16 - Path of Meditation (Dhyanyoga)'.] The one who enlightens on these states is Chintamani. The theory according to the Mudgal Puran is that by devotion unto Chintamani, the five states of the subconscious mind are destroyed and total Serenity (Shanti) is acquired.'
'Man' means complete and 'glu-gayate' means one endowing Serenity or purity. That which purifies both internally as well as externally is auspicious (mangal). An icon bringing auspiciousness is Mangalmurti.
In Maharashtra, 'Mangalmurti morya' is used to proclaim Ganapati's glory. The word 'morya' in it refers to a famous devotee of Lord Ganesh from the fourteenth century, Morya Gosavi from ChinchvaD, near Pune in Maharashtra. This depicts the inseparable relationship between God and His devotee.
Uma is Parvati and phal means product. Ganapati acquires this Name as He is the product, that is the son of Parvati. Umaphal also means spiritual knowledge (dnyan). Ganapati is the deity of spiritual knowledge. So this Name befits Him in both ways.
Lord Ganesh is the master of the 18 vidyas - 1. Science of proper articulation and pronunciation (shiksha), 2. A part of the Vedas explaining rituals (kalpa), 3. Grammar (vyakaran), 4. Etymological explanation of difficult Vedic words (nirukta), 5. Astrology (jyotish), 6. Science of prosody (chandas), 7. Rugveda, 8. Yajurveda, 9. Samaveda, 10. Atharvaveda, 11. Purva-Uttarmimansa (concerned with the correct interpretation of Vedic rituals and the settlement of dubious points with regard to Vedic texts), 12. System of Hindu philosophy founded by Sage Gautam (nyaya), 13. Purans, 14. Science of Righteousness (Dharmashastra), 15. Ayurveda, 16. Dhanurveda, 17. Gandharvaveda and 18. Science of ethics or morality (nitishastra) Hence before commencing the study of any of these or in the ritual performed for their study, the ritualistic worship of Lord Ganesh is important.
The Vedas are referred to as Brahman. They may also be termed as Vedabrahman. Lord Ganesh is the master of the mantras in these Vedas. Hence He is called Brahmanspati.
One finds that Lord Ganesh is known by varied names in various regions. The Dravidian culture calls Lord Ganesh 'Rajmukh' or 'Murugan'. In Nepal, He is known as 'Suryaganapati', in Myanmar (formerly Burma) as 'Mahapini', in Mongolia as 'Dhotkar', in Tibet as 'Ekadanta', in Cambodia as 'Pradganesh', in the Java islands as 'Kalantak', in China as 'Kvanshitiyik' while in Japan He is known as 'Vinayakasha'.
Reference: 'Ganapati', Published by Sanatan Sanstha