avyakto ‘yam acintyo ‘yam
avikāryo ‘yam ucyate
tasmād evaḿ viditvainaḿ
nānuśocitum arhasi

Translation of Bhagavad Gita 2.25

It is said that the soul is invisible, inconceivable and immutable. Knowing this, you should not grieve for the body.

Commentary by Sri A.C. Bhaktivedanta Swami Prabhupada of Gaudiya Sampradaya:

As described previously, the magnitude of the soul is so small for our material calculation that he cannot be seen even by the most powerful microscope; therefore, he is invisible. As far as the soul’s existence is concerned, no one can establish his existence experimentally beyond the proof of shruti, or Vedic wisdom. We have to accept this truth, because there is no other source of understanding the existence of the soul, although it is a fact by perception. There are many things we have to accept solely on grounds of superior authority. No one can deny the existence of his father, based upon the authority of his mother. There is no source of understanding the identity of the father except by the authority of the mother. Similarly, there is no source of understanding the soul except by studying the Vedas. In other words, the soul is inconceivable by human experimental knowledge. The soul is consciousness and conscious—that also is the statement of the Vedas, and we have to accept that. Unlike the bodily changes, there is no change in the soul. As eternally unchangeable, the soul remains atomic in comparison to the infinite Supreme Soul. The Supreme Soul is infinite, and the atomic soul is infinitesimal. Therefore, the infinitesimal soul, being unchangeable, can never become equal to the infinite soul, or the Supreme Personality of Godhead. This concept is repeated in the Vedas in different ways just to confirm the stability of the conception of the soul. Repetition of something is necessary in order that we understand the matter thoroughly, without error.Commentary by Sri Vishvanatha Chakravarthi Thakur of Gaudiya Sampradaya:

(combined commentary for verses 24 and 25)

Therefore the atma is described in this way (taking the descriptions of the previous verse): it cannot be cut, burned, dried or moistened. The repetition of the word signifying the eternal nature of the soul in this section (such as the use of the words nitya and sanatanah in this verse, and statements of previous verses) is for the purpose of precisely defining the soul for those of confused intellect, just as one will repeat several times in order to make certain that everyone understands, “This is the dharma of kali yuga, this is the dharma of kali yuga.”

Sarva gatah (gone everywhere) means that the soul goes into all sorts of bodies such as human, animal and deva by its karma. Sthanu and acala have the same meaning, “having stability,” and the repetition is to make the meaning clear.

Because it is very fine, it is called avyakta. Because it pervades the body with consciousness, it is called acintya, beyond conception. Being devoid of the six changes such as birth, it is called avikarya.Commentary by Sri Ramanuja of Sri Sampradaya:

2.25 The self is not made manifest by those Pramanas (means of knowledge) by which objects susceptible of being cleft etc., are made manifest; hence It is unmanifest, being different in kind from objects susceptible to cleaving etc., It is inconceivable, being different in kind from all objects. As It does not possess the essential nature of any of them. It cannot even be conceived. Therefore, It is unchanging, incapable of modifications. So knowing this self to be possessed of the above mentioned qualities, it does not become you to feel grief for Its sake.
Commentary by Sri Sridhara Swami of Rudra Sampradaya:

Analysing the subject further it is known as avyaktah or unmanifest not being able to be perceived by the eye or any other of the senses, acintyah or inconceivable not being able to be perceived by the mind. The word avikaryah meaning unchangeable is in the sense of not being modifiable by any of the organs of action. The word ucyate meaning it is said thus indicates that authoritative testimony corroborates the validity of the immortal soul being eternal and everlasting as given in the previous verse. The word tasmat means therefore and by its use it is shown that this is a concluding statement.
Commentary by Sri Madhvacharya of Brahma Sampradaya:

he Supreme Lord is verily the manifestation of the inconceivable Himself.

Now begins the summation.

It might be questioned that if the Supreme Lord is all pervading why is He not visible? The reason is because He is inconceivable. It may be further queried how does He appear so endowed? Because of His power of remaining unthinkable; but this power does not come from any outside source but is contained solely within Himself. Whatever forms He exhibits in all these He alone manifests fully. To emphasise words like enam and ayam indicating the eternal soul are used in different contexts. In the case of the living entities it is to show that their eternal soul has qualitatively the same essence and attributes as the Supreme Lord but infinitesimal quantitatively. This is further confirmed in various Vedic scriptures.

In the Paingi Struti it is stated:

The physical body, pleasures and fragrances, light of wisdom, impeccable bravery; these things reflect the best of all wisdom, the best of all pleasures and bravery which find their source within the Ultimate Supreme Personality the Supreme Lord Krishna.

In the Parama Sruti it is stated:

The Supreme Lord Krishna is known to be both possessing a body and devoid of body. This is because He possesses a spiritual transcendental body but he is not possessing a physical body. Because His body is not constituted from the elements of material nature it is said to be a-dehah. The head, the feet, the arms and other parts of this spiritual, transcendental body are made up of the Supreme Lord Himself. There exists nothing which is distinctive from this spiritual, transcendental nature which can be called His body therefore He is called a-dehah without body. He Himself is His form and this spiritual, transcendental form is eternally existing beyond the scope of material existence.

Other than the Supreme Lord Krishna, none else is competent to comprehend this form. There is never for the Supreme Lord the creation or separation or destruction of His spiritual, transcendental body. He and His body are one and the same spiritual, transcendental essence. All of His beauty and all of His wonderful and phenomenal attributes are recorded only to understand that He is the source, the original repository of all attributes. The awareness that I am this physical body is an experience common to all living entities. Even though such a distinct awareness is normal it is perceived as special. Even though it is special, one does not perceive it as being separate from one’s own individual consciousness. So how can there ever be a manifestation of the resplendent Supreme Lord for one who cannot even first perceive their own individual consciousness within themselves?

Although attributes may which appear contradictory in normal, mundane human behaviour, it should be understood without a doubt that no such discrepancy ever exists in regard to the Supreme Lord when spoken of in various scriptures. Discrepancies imagined during reflection or speculated due to non-reflection are all a product of ignorance. They do not exist as a reality. These attributes are self revealed.

In the Suparna section of the Rig Veda it is stated:

The one only, without a second, before which there was nothing else whatsoever. Death was enveloped by death. Nothing else was visible. Like the waters which emanate from the mountains flow down in various rivers, whoever sees the attributes of the Supreme Lord in various manifested forms loses merit. Whoever sees even the slightest difference in the descents of the Supreme Lord Krishna’s multifarious and variegated incarnations such as Narasingha, Kurma and others from His attributes and actions and also a difference in these manifestations will go to the world of nescience. Therefore living entities who desire to learn about the Supreme Lord should start by knowing Him as indivisible.

Even if there appears to be some special difference it should be understood to be so due to time and circumstance. For example there is no difference between His special attributes and His special strength. There is as well no factual difference whatsoever between any special attribute and any of His special incarnation and His supreme distinctive and indivisible aspect. There is complete and absolute continuity and similarity in all respects in His original two armed form as the Supreme Lord Krishna and all of His unlimited incarnations and expansions.

There are six categories of incarnations of the Supreme Lord Krishna. The Four Kumaras, Hamsa, Varaha, Narada, Nara-Narayana, Kapila, Yagna, Dattatreya, Hayagriva, Prsnigarbha, Rsabha, Prithu, Matsya, Kurma, Dhanvantari, Mohini, Narasimha, Vamana, Parasurama, Vyasa, Rama, Balarama, Buddha and Kalki are known as Lila Avatars. The three manifestations of Visnu are known as Purusa Avataras of the Supreme Lord Krishna. Brahma and Shiva are known as Guna Avatars, incarnations such as Hari and Ajita are known as Manvantara Avatars, incarnations such as Anantadeva and Sesha Naga are known as Saktyavesa Avatars and incarnations such as Sukla, Rakta, Shyama and Pita are known as Yuga Avatars.

In the Varaha Purana it is stated:

All these avatars are bonafide incarnations of the Supreme Lord Krishna and then there are still all of His numerous, authorised expansions and His unlimited associates who descend to a material world with Him to assist in His pastimes. Those who have been blessed to know the Supreme Lord in this manner, onto them is the love of the Supreme Lord Krishna bestowed. Those who think otherwise due to misguided false conceptions, lack of opportunity for learning the Ultimate Truth or the inherent limitations of their own mind will fall permanently into the worlds of nescience.
Commentary by Sri Keshava Kashmiri of Kumara Sampradaya:

In this verse Lord Krishna gives additional reasons why one should never grieve for the soul. Because the soul is unmanifest it is not able to be cut or burned like objects which are manifest in this world and being unmanifest it is inconceivable as it is not possible to equate it with the form or nature of anything that one could imagine in the material existence. Being unable to imagine the constitution of the soul indicates it is immutable because it is devoid of any process of modification for example milk transforming to youghurt. Therfore comprehending the nature and quality of the soul as enunciated above one should not be under the delusion of lamentation.
Commentary by Sri Adi Shankaracharya of Advaita Sampradaya:

2.25 Moreover, ucyate, it is said that; ayam, This, the Self; is avyaktah, unmanifest, since, being beyond the ken of all the organs, It cannot be objectified. For this very reason, ayam, This; is acintyah, inconceivable. For anything that comes within the purview of the organs becomes the object of thought. But this Self is inconceivable becuase It is not an object of the organs. Hence, indeed, It is avikaryah, unchangeable. This Self does not change as milk does when mixed with curd, a curdling medium, etc. And It is chnageless owing to partlessness, for it is not seen that any non-composite thing is changeful. Not being subject to transformation, It is said to be changeless. Tasmat, therefore; vidivata, having known; enam, this one, the Self; evam, thus, as described; na arhasi, you ought not; anusocitum, to grieve, thinking, ‘I am the slayer of these; these are killed by me.’
Commentary by Sri Abhinavagupta of Kaula Tantra Sampradaya:    

2.23-25 Nainam etc. upto arhasi. The weapons etc., that cause destruction, haldly do anything to This. For, being, by nature, exclusively pure Consciousness, remaining without support, having no component parts and being independent, this cannot be destroyed through the process of either assumption of an altogether different nature, or the destruction of the support, or the mutual separation of the component parts, or the rise of an opponent, and so on. Nor the act to going to another body is a new thing for This. For, even when This is [apparently] with a single body, This travels always to different body; for the body does not remain the same even for a moment. By understanding this Self to be as such, you should not lament This.
Sanskrit Shloka Without Transliteration Marks:

avyakto ‘yam acintyo ‘yam
avikaryo ‘yam ucyate
tasmad evam viditvainam
nanusocitum arhasi
Sanskrit to English Word for Word Meanings:

acchedyaḥ — unbreakable; ayam — this soul; adāhyaḥ — unable to be burned; ayam — this soul; akledyaḥ — insoluble; aśoṣyaḥ — not able to be dried; eva — certainly; ca — and; nityaḥ — everlasting; sarva-gataḥ — all-pervading; sthāṇuḥ — unchangeable; acalaḥ — immovable; ayam — this soul; sanātanaḥ — eternally the same.