What is ātman?

From the second chapter “The eternal reality of soul’s immortality” - सांख्ये बुद्धिर्योग
य एनं वेत्ति हन्तारं यश्चैनं मन्यते हतम्
उभौ तौ न विजानीतो नायं हन्ति न हन्यते (2.19)

yaḥ enaṁ vetti hantāram yaś cainaṁ manyate hatam
ubhau tau na vijānīto nāyaṁ hanti na hanyate (2.19)

yaḥ - Anyone who; enaṁ - the soul; vetti – thinks; hantāram-is the slayer; yaś cainaṁ manyate - anyone who thinks the soul; hatam – slain;
tau ubhau – both of them; na vijānīto – are in ignorance; nāyaṁ - the soul never; hanti na hanyate – slays or is slain

He holds the Atman as slayer and he considers it as slain, both of them are ignorant. It slays not, nor is it slain.
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न जायते म्रियते वा कदाचित् नायं भूत्वा भविता वा न भूयः
अजो नित्य: शाश्वतोऽयं पुराणो न हन्यते हन्यमाने शरीरे (2.20)

na jāyate mriyate vā kadāchit nāyaṁ bhūtvā bhavitā vā na bhūyaḥ
ajo nityaḥ śāśvato ‘yaṁ purāṇo na hanyate hanyamāne śarīre (2.20)

na jāyate – is never born; mriyate vā kadāchit – neither never dies; āyaṁ - soul; bhūtvā – nor does it come into being; bhavitā vā – when the body is created; na bhūyaḥ - again
ajo –birthless; nityaḥ-eternal; śāśvato‘yaṁ - imperishable; purāṇo- timeless; na hanyate – is never destroyed; hanyamāne śarīre – when the body is destroyed

The soul (atman) is neither born nor does it die. Nor does it come into being again when the body is created. The soul is birthless, eternal, imperishable and ancient. It is not killed when the body is slain.

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वासांसि जीर्णानि यथा विहाय नवानि गृह्णाति नरोऽपराणि
तथा शरीराणि विहाय जीर्णानि अन्यानि संयाति नवानि देही (2.22)

vāsāṁsi jīrṇāni yathā vihāya navāni gṛhṇāti naro ‘ parāṇi
tathā śarīrāṇi vihāya jīrṇāni anyāni saṁyāti navāni dehī (2.22)

vāsāṁsi – garments; jīrṇāni – old worn out; yathā – just as; vihāya – giving up; navāni – new; gṛhṇāti – accepts; naro – man; ‘ parāṇi – other
tathā – similarly; śarīrāṇi – bodies; vihāya – giving up; jīrṇāni –old worn out; anyāni – other; saṁyāti – accepts; navāni – new; dehī – embodied soul

Just as man giving up old worn out clothes accepts other new garment; similarly the embodied soul giving up old worn out bodies very accepts new body
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नैनं छिन्दन्ति शस्त्राणि नैनं दहति पावकः
न चैनं क्लेदयन्त्यापः न शोषयति मारुतः (2.23)

nainaṁ chindanti śastrāṇi nainaṁ dahati pāvakaḥ
na cainaṁ kledayantyāpaḥ na śoṣyati mārutaḥ (2.23)

na + enaṁ - the soul never; chindanti – harm; śastrāṇi – weapons; nainaṁ; dahati – burn; pāvakaḥ - fire;
na cainaṁ; kledayantyāpaḥ - wet by water; na; śoṣyati – dry; mārutaḥ - air

Weapons do not cleave the atma; fire burns it not, water wets it not, wind dries it not.
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अच्छेद्योऽयमदाह्योयं अक्लेद्योऽशोष्य एव च |
नित्यः सर्वगतः स्थाणुः अचलोऽयं सनातनः|| (2.24)

acchedyo ‘yam adāhyo ‘yam akledyo ‘śoṣya eva ca
nityaḥ sarvagataḥ sthāṇur acalo ‘yam sanātanaḥ (2.24)

acchedyo ‘yam – soul is uncleavable; adāhyo ‘yam –soul can’t be burn; akledyo – insoluble; ‘śoṣya eva ca - unwitherable
nityaḥ - eternal; sarvagataḥ - all pervading; sthāṇur – stable; acalo - immovable ‘yam sanātanaḥ - everlasting

The soul is indestructible, the soul is incombustible, neither wetted, nor dried. The soul is eternal, all pervading, unmodified, immovable and everlasting.
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अवक्तोयमचिन्त्योऽयं अविकार्योऽयमुच्यते (2.25)
avakto ‘yam acintyo‘yam avikāryo‘yam ucyate (2.25)

avakto - imperceptible; ‘yam; acintyo‘yam – soul is inconceivable; avikāryo ‘yam – soul is immutable; ucyate – it is declared that the soul;

The atman is said to be imperceptible, inconceivable, immutable.
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TBD
What is the difference
1. Between jiva and atma?
2. Between jivatma and paramatma?
3. Between Atma and brahman?
4. Between Brahman and Paramatma?
5. Between God and Brahman?