Q:

I want to know the exact explanation of Maha Mrityunjaya Mantra

Can any one give the exact explanation of this mantra? What is the logic behind chanting this Mantra?
How the mind of a person related to this and is it true that we can save the life of a dying person if we chant this mantra for him?

A:

The Mahamrityunjaya Mantra 
In Devanagari script:

ॐ त्र्यम्बकं यजामहे सुगन्धिंम् पुष्टिवर्धनम् ।
  उर्वारुकमिव बन्धनान् मृत्योर्मुक्षीय माऽमृतात् ।।

In English script:

om tryambakaṃ yajāmahe sugandhiṃ pushtivardhanam ǀ
  urvārukamiva bandhanān mrityormukshīya mā'mritāt ǁ

This is Great Death-conquering Mantra, also called the Tryambakam Mantra, is a verse of the Rigveda. It is addressed to Tryambakam, "the three-eyed one", Lord Shiva. It is also called the Rudra mantra, referring to the furious aspect of Lord Shiva. It is sometimes known as the Mrita-Sanjivini mantra because of its "life-restoring" powers.
tryambakaṃ (त्र्यम्बकं ) - Three-eyed(Knowledge of future,Past & Present) God, Shiva
yajāmahe (यजामहे ) - honour
sugandhiṃ (सुगन्धिंम् ) - sweet smelling
pushti (पुष्टि) - prosperous
vardhanam (वर्धनम्) - strengthening
urvārukamiva (उर्वारुकमिव )  - like the cucumber
bandhanān (बन्धनान् ) - tied down (under captivity)
mrityor (मृत्यो) - from death
mukshīya (र्मुक्षीय) - liberate, free
mā'mritāt (माऽमृतात्) - give me immortality
Meaning: We worship the Three-eyed Lord who is fragrant and who nourishes and nurtures all beings. As is the ripened cucumber freed from its bondage (to the creeper), may He liberate us from death for the sake of immortality.
Maha Mrityunjaya Mantra is the great mantra for conquering death for it protects against all threats with its divine vibrations. It purifies the karmas of the soul at a deep level. It is beneficial for mental, emotional and physical health. It is also a moksha mantra which bestows longevity and immortality.

A:

MahaMrityunjaya mantra is one of the most special mantra from Veda Samhita addressed to Lord Shiva.

It is in Rig Veda Samhita 7.59.12. Sage Vasistha is the Rishi (who heard) of the mantra and Devata is 'Rudra'. Similarly Mahamrityunjaya mantra is also avaliable in YajurVeda Samhitas like Taittariya Samhita 1.8.6.i, Vajasaneyi Samhita in 3.60 etc...

The mantra is as:

त्र्यम्बकं यजामहे सुगन्धिं पुष्टिवर्धनम्
उर्वारुकमिव बन्धनान्मृत्योर्मुक्षीय माऽमृतात् ।।

om tryambakaṃ yajāmahe sugandhiṃ pushtivardhanam ǀ
urvārukamiva bandhanān mrityormukshīya mā'mritāt ǁ

The meaning of mantra is as:

We worship the Three-eyed Lord who is fragrant and who nourishes and nurtures all beings. As is the ripened cucumber freed from its bondage (to the creeper), may He liberate us from death for the sake of immortality.

Sukra while teaching this mantra to Dadhichi tells that this is the best mantra as per him:

मृतसञ्जीवनीमन्त्रो मम सर्वोत्तमः स्मृतः ।
एवं जपपरः प्रीत्या नियमेन शिवं स्मरन् ।। (Shiva Purana Rudra Samhita chapter 38)  
In my opinion, this Mrityunjaya mantra is the excellent of all. Repeat the mantra regularly with your mind devoted to Shiva.

Lord Trayambakeshwara appears before Sage Dadhichi as he does penance by this mantra and he asks for three boons:

दधीच उवाच
देवदेव महादेव मह्यं देहि वरत्रयम् ।
वज्रास्थित्वमवध्यत्वमदीनत्वं हि सर्वतः ।। 
Dadhicha said "O Great lord of the gods, kindly bestow three boons to me. First my bones should be strong like Vajra, No one should be able to kill me and I should never feel distress.

Similarly Suta also states that Mahamrityunjaya mantra is the best of all Veda mantras:

नानेन सदृशो मंत्रो लोके वेदे च सुव्रताः ।
तस्मात्त्रियंबकं देवं तेन नित्यं प्रपूजयेत् ।। (Linga Purana II.54) 
There is no mantra better in the world as well as in Vedas (than this). Therefore one should always adore the God Trayambakeswara in the world.

Explanation of the words of the mantra is avaliable in Shiva Purana and Linga Purana;
Tryambakam:
Trayambakam denotes that Lord who has three eyes is the lord of three worlds, three Gunas, Trimurtis etc.. or Lord of everything which exists in Triad:

त्रयम्बकं यजामहे च त्रैलोक्यपितरं प्रभुम् ।
त्रिमण्डलस्य पितरं त्रिगुणस्य महेश्वरम् ।। (Shiva Purana Rudra Samhita chapter 38)

We worship lord Shiva having three eyes, who is the lord of three worlds and the master of three gunas.

त्रयाणामपि लोकानां गुणानामपि यः प्रभुः ।
वेदानामपि देवानां ब्रह्मक्षत्रविशामपि ।
अकारोकारमकाराणां मात्राणामपि वाचक ।।
तथा सोमस्य सूर्यस्य वह्नेरग्नित्रयस्य च ।
अंबा उमा महादेवो ह्यंबकस्तु त्रियंबक ।। (Linga Purana II.54)

The word ‘Tryambaka’ has been defined as the one, who happens to be the lord of the three worlds, three gunas, three Vedas, three gods, three castes— He is revealed by the three syllables ‘a’, ‘u’, ‘m’. He is the lord of three fires, Viz. : the sun, the moon and the fire. He represents Amba, Uma and Mahadeva. Thus, the lord of all the sets of three is known as Tryambaka.

Yajāmahe:
Mantra contains the words 'Yajãmahe' to denote that such Lord who is Trayambaka is to be worshipped by various methods:

तं पुष्टीवर्धनम् देवं घृतेन पयसा तथा ।
मधुना यवगोधुममाषबिल्वफलेन च ।।
कुमुदार्कशमीपत्रगौरसर्षपशालिभिः ।
हुत्वा लिंगे यथान्यायं भक्त्या देवं यजामहे ।। (Linga Purana II.54) 
Yajãmahe (we worship him) is thus defined. We worship lord Siva in a proper manner and devoutly by performing homas with ghee, milk, honey, wheat, black gram, wood apple fruits, lilies, Arka flowers, Sami leaves, white mustard and Sali grains in a Linga.

Sugandhim:
Lord is called 'Sugandhim' because he is the essence of Everything which exists in this Universe just as Sugandha (fragrance) is the essence of a flower:

इन्द्रियेषु तथान्येषु देवेषु च गणेषु च ।
पुष्पे सुगन्धिवत्सरः सुगन्धिरमरेश्वर ।। (Shiva Purana Rudra Samhita chapter 38) 
As the fragrance is enshrined in the flowers, similarly he is found present in all the sense organs, the gods, gunas, in a very strong manner. And thus he is the Ishwara.

सुपुष्तितस्य वृक्षस्य यथा गंधः सुशोभनः ।
वाति दूरात्तथा तस्य गंधः शंभोर्महात्मन ।।
तस्मात्सुगन्धो भगवान्गंधारयति शंकर ।
गांधारश्च महादेवो देवानामपि लीलया ।।
सुगंधस्तस्य लोकेस्मिन्वायुर्वाति नभस्तले ।
तस्मात्सुगन्धिस्तं देवं सुगंधी पुष्टिवर्धनम् ।। (Linga Purana chapter 54) 
The splendid fragrance of a tree in a full bloom is wafted to a great distance. Similar is the fragrance of the great Lord. The lord is the fragrance or the holder of the fragrance on all other Devas. His fragrance is wafted in the world where the wind blows. Hence the attribute Sugandhim refers to the Lord. Now the word Pushtivardhanam is being defined.

Pushtivardhanam:
Lord is called Pushtivardhanam because he is the cause of the growth of the Universe, and source of nourishment of everyone:

विष्णोः पितामहस्यापि मुनीनां च महामुने ।
इन्द्रियाणां च देवानांतस्माद्वै पुष्टिवर्द्धनः ।। (Shiva Purana Rudra Samhita chapter 38) 
O Excellent sage, he happens to be Brahma, Visnu and other gods, who derive their nourishment from him. 

यस्य रेतः पुरा शम्भोहरेर्योनौ प्रतिष्ठितम् ।
तस्य वीर्यादभूदण्डं हिरण्यमयमजोद्भवम् ।।
चंद्रादित्यौ सनक्षत्रौ भूर्भुवः स्वर्महस्तपः ।
सत्यलोकमतिक्रम्य पुष्टिवीर्यस्य तस्य वै ।।
पञ्चभूतान्यहंकारो बुद्धिः प्रकृतिरेव च ।
पुष्टिर्बीजस्य तस्यैव तस्माद्वै पुष्टिवर्धनः ।। (Linga Purana II.54)

Formerly, the semen of the lord Shiva was deposited in the womb of Vishnu. It was from this semen that the cosmic golden egg, the source of origin of Brahma originated. The nourishment of his semen extends beyond the moon, sun, stars, earth, Bhuvah. Svah. Mahah, Tapah and Satya. The Five elements, the cosmic ego. the cosmic intellect and Prakrti are nourished by his semen virile Hence. the lord is Pushtivardhanam (the nourisher).

urvārukamiva bandhanān mrityormukshīya mā'mritāt ǁ
It is the latter half part of the Mantra. In the former part, Lord is praised with various attributes while this latter half part asks from the Lord. Mantra says that just as the ripened cucumber is freed from it's bandhan (bond) in the similar way may Lord free us from death and give Moksha which is Immortal.

सत्येनान्येन सूक्ष्मायान्मृत्युपाशाद्भवः स्वयम् ।
बन्धमोक्षकरो यस्मादुर्वारुकमिव प्रभुः।। (Shiva Purana Rudra Samhita, chapter 38) 
you will be surely freed from the noose of Yama. The lord is the cause of both bondage and Moksha and hence Lord is denoted as Urvarukamiva (in the mantra). 

ऋतेनानेन मां पाशाद्वंधनात्कर्मयोगतः ।
मृत्योश्च बन्धनाच्चैव मुक्षीय भव तेजसा ।।
उर्वारुकाणां पक्वानां यथा कालादभेत्पुनः ।
तथैव कालः संप्राप्तो मनुना तेन यत्नतः ।। (Linga Purana II.54) 
The later half of the mantra is now interpreted. By the grace of the Rta (systematic worship), let me be released from the bondage of actions and reactions as well as fetters of death. Let me be released from the resplendence of the worldly existence. Just as a mature cucumber falls from it's stalk when the season arrives, so also, the time of liberation has arrived by the grace of mantra.

Thus it is very clear that the Mantra is primarily asking Moksha from Lord Shiva so that one can become Immortal as Moksha is related to Immortality. But the Mantra is also chanted  to gain long life in the worldly existence. It is because as clear from the above, Lord becomes the cause of bondage and liberation for the chanter. And one is also freed from the Bandhan of Kãla (i.e., Yama Pãsha) as he is freed from every kind of Pãshas:

शिवस्य सन्निधौ ध्यात्वा नास्ति मृत्युभयं क्वचित् ।। (Shiva Purana Rudra Samhita chapter 38) 
In case the penance is done in presence of Shiva, there is no fear of death from any quarters.

एवं मन्त्रविधिं ज्ञात्वा शिवलिंगं समर्चयेत् ।
तस्य पाशक्षयोऽतीव योगिनो मृत्युनिग्रहः ।।  (Linga Purana II.54)

One should worship ShivaLinga after well understanding the meaning and the rites related to Mantra. The Yogi shall severe all the bonds. He can control death.

