Achintya Bheda Abheda tattva means inconceivable oneness and difference.

Understanding "Achintya Bheda Abheda tattva" which means-

"Inconceivable oneness and difference."

1 - achintya means inconceivable.

2 - bheda means difference.

3 - abheda means non-difference.

4 - tattva means principle, reality  truth, absolute truth, knowledge, that-ness (the condition of being an existent thing apart from whatever may be known or stated about that thing), real nature.

Lord Caitanya Mahaprabhu (1486–1534), the founder of the philosophy  of Achintya Bheda Abheda tattva school and Gaudiya Vaisnavism.

The Gaudiya Vaisnava religious tradition employs the term in relation to the relationship of creation and creator (Krsna, Svayam Bhagavan), between God and his energies (sakti).

This philosophy was taught by Sri Caitanya Mahaprabhu and differentiates the Gaudiya tradition from the other Vaisnava Sampradayas.

It can be understood as an integration of the strict dualist (dvaita) theology of Madhvacharya and the qualified monism (vishishtadvaita) of Ramanuja.

Historically within the Vedic teachings, there are two conflicting philosophies regarding the relationship between the individual marginal living entities (also known as jiva-tattva, jiva-soul jivatma) and Lord Krsna (God) the Supreme Personality of Godhead and cause of all causes,  that includes His Visnu-tattva expansions (Narayana or Visnu forms)

There are four categories of different living entities- 

1 - Jiva-tattva, 78.125% of Krsna's quantities having 50 of Krsna's 64 attributes.  

2 - Visnu-tattva, 93.75 of Krsna's qualities having 60 of Krsna's 64 attributes.

3 - Visnu-sakti-tattva, (internal potency sakti refering to Visnu-tattva expansions) 93.75 of Krsna's qualities having 60 of Krsna's 64 attributes.

4 - Siva-tattva, 84.375 of Krsna's qualities having 55 of Krsna's 64 attributes.

Advaita schools assert the monistic view that the individual jiva-soul and God are one and the same, whereas Dvaita schools give the dualistic argument that the individual jiva-soul and God are eternally separate.

The philosophy of Achintya-bheda-abheda tattva includes elements of both viewpoints. 

The individual jiva-soul is intrinsically linked with the Supreme Lord, and yet at the same time is not the same as God, each have their independent personality.

The exact nature of this relationship is inconceivable to the human mind.

All individual jiva-souls are considered to be part and parcel of the Supreme Lord like the sun-rays and the sun-disc are joined together and are inseparable.

This means the individual jiva-souls are the same in quality but not in quantity to Krsna.

As said above, the individual jiva-souls only have partial features of Krsna's qualities which is 50 of Krsna's 64 qualities, or 78.125% of Krsna's opulences.

Another analogy in this context is God or Krsna is compared to a fire and the jiva-souls are compared to sparks coming off of the flame.

The theological tenet of achintya-bheda-abheda tattva reconciles the mystery that Krsna is simultaneously one with and different from His creation.

In this sense, Vaisnava theology is panentheistic as in no way does it deny the separate existence of God (Krsna and His Visnu/Narayana expansions)

However, at the same time, creation (or what is termed in Vaisnava theology as the cosmic manifestation) is never separated from Krsna.

Krsna as His different expansions, always exercises supreme control over his creation, sometimes directly, but most of the time indirectly through his different potencies or energies (Prakrti)

Examples are given of a spider and its web; earth and plants that come forth and hair on the body of human being.

Srila Prabhupada - "One who knows God (Krsna) knows that the impersonal conception and personal conception are simultaneously present in everything and that there is no contradiction. Therefore Lord Caitanya established His sublime doctrine- 

"Acintya bheda-and-abheda-tattva means simultaneously one and different." (Vaniquotes sastric quotes)

The analogy often used as an explanation in this context in the relationship between the Sun-disc (compared to Krsna) and the sun-rays (compared to sakti or the energies of Krsna) mentioned above.

For example, both the sun-disc and sun-rays can not exist without each other and can only exist together.

But there is also another difference between between the Sun-disc and the sun-rays.

There is a difference to having a beam of sunshine in your room, and being in close proximity to the sun-disc itself.

Qualitatively the Sun-disc and the sun-rays are not different, but as quantities they are very different.

This analogy is applied to the individual jiva-souls and Krsna, the jiva-souls being of a similar quality to Krsna, but not sharing the same qualities to an infinite extent, as would the Supreme Personality of Godhead himself has.

Thus there is a difference between the jiva-souls and the Supreme Personality of Godhead Krsna.

What is the difference in concept to Advaita Vedanta?

It is clearly distinguished from the concept of anirvacaniya (inexpressible) of Advaita Vedanta. 

There is a clear difference between the two concepts as the two ideas arise for different reasons.

Advaita concept is related to the ontological status of the world, whereas both Svayam bhagavan and his saktis (in Krsna Himself and his intimate powers) are empirically real, and they are different from each other, but at the same time they are the same. 

But that does not negate the reality of both.

There are exceptions-

While it applied to relations between Purusha (the Lord) and Prakriti (be it material, marginal, or spiritual powers), in the theology of the concept there are areas of exceptions.

Jiva Goswami also accepts that any object and its energy are non-different, such as fire and power of burning. 

While some maintain that its only a secondary extension of the principle that it is primarily applied to Svayam bhagavan Krsna and His energies.

It does not, however, apply to differences between Avatars of Svayam bhagavan and Lord Himself, so the difference between Krsna and His origin, is not covered by the concept of acintya bhedabheda tattva, i.e. it cannot be applied in cases where different levels of Purusha are compared.

This is a very difficult subject to understand and requires strictly chanting at least 16 rounds of the Hare Krsna mantra daily, and engaging in devotional service.

Such voluntary surrender to a bonafide genuine Spiritual Master will eventually purify us and allow us all to understand what is written here, there is no other way.

None devotee academics, scholars, show bottle gurus and impersonalists will never understand the real teachings of Lord Caitanya Mahaprabhu..**..