Consider a sheet of paper, when you crumple it, that is Ahankar (ego). As you slowly open the sheet and try to lay it flat as it used to be, that form is called Aatma. 

In Yoga, they refer to this concept as Chitta Vritti Nirodha.
If you take a string and knot it at several locations (Ahankar), the process of removing each knot is through Gyan (knowledge) and the knot-free string thus obtained, is the Aatma. One attains it through Aatma-gyan.

So the spectrum of Aham (one's identity) and Aatma (Consciousness), is such that you move from Ahankar (ego) towards Aatma, upon obtaining Aatma-gyan (self-realization).

The spectrum of Aham and Aatma

Aatma doesn't exist in Buddhism.
Buddha preached that nothing lasts forever. Everything is temporary and everything should meet its end.
What one refers to Aatma, is in fact a reference to self, to one's identity and thus one's ego that has been carefully nourished over the years.

State University of New York at Buffalo

Some refer to it as the soul, but when you dig deeper into its philosophy, it is refers to Consciousness. 
One is differentiated as living from the non-living when this consciousness awakens the hunger within the being. This consciousness awakens ones senses, thus allowing them to partake in this world and its associated materialistic life. 


Aatma

Pattanaik, Devdutt (2017), Devlok with Devdutt Pattanaik, Penguin Random House Publishing

Devlok with Devdutt Pattanaik

That which is Nirgun (has no form), Anaadi (from the beginning), Ananta (endless), specifically, that which is not bound by the concept of time, that has always existed and will continue to exist is called Aatma.

Etymology of Aatma

To understand the concept of Aatma, a poem was written in the Rig Veda, about 8500 years ago. 

There exist two birds, seated on a branch on a tree. One satiates itself by eating a fruit, while the other watches the previous eating the said fruit. 

This is a metaphor. The bird that is eating the said fruit is referencing one's body, that partakes in the intricacies of life, and uses its sense organs to enjoy every aspect of this life, while the bird watching the other is one's Aatma. 

There are three levels to this philosophy. 

One states that "I am observing you".

The other states that my Mann, bird 2 (mind) watches over my Shareer, bird 1 (body). In other words, "I am observing myself".

The third questions, "What entity watches over my mind?"

So the third level states that my Aatma (consciousness) watches over my Mann (mind), or "I am observing my own mind". In English, we may refer to it as recursion. This ability exists only in humans. This trait of observing oneself, as a witness, without getting emotionally involved, stems from the existence of Aatma. 

Rig Veda, along with Upanishads, also state that humans that search for Artha (meaning) to one's life, is moving towards Aatma-gyan (self-realization).

Vedic Philosophy of Aatma

-Brahman, Param-aatma

-Jeev-aatma

-Para-aatma

Forms of Aatma

Aatma is referred to as Dravya, that which awakens your senses. It is that entity that carries the weight of Karma on its back.

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