Read more about Equanimity from the works of the Mother and Sri Aurobindo.
What is Equanimity?
Equanimity or samata has been described as the capacity to retain inner calmness in all situations, pleasant or unpleasant. It is an absolutely essential quality to cultivate in oneself to sustain on the path of yoga that has many ups and downs. ‘Equality of the soul’ is the psychological capacity to face these without becoming overjoyed or discouraged. There is also an equality in the body, a physical poise that supports this state; and a mental poise, of being able to see that all thoughts and ideas, no matter how contradictory as having their place in the larger scheme of things. This equality comes from a deep sense of unity with the universe that enables us to perceive the divine will in all things. That is not to say that one must become indifferent or ignorant. It is possible to acknowledge pleasantness and unpleasantness without becoming identified with it and remaining still in our core selves. This can be better understood through the two types of equanimity:   
Active & Passive Equanimity
Passive Equanimity: This refers to the aforementioned sense of faith in the divine will which enables one to carry on with a passive endurance and equal acceptance of all occurrences along the way. It assists the individual in overcoming duality and conquering the lower movements to pave the way for a more active equanimity to manifest in us.
Active Equanimity: This comes from more than just faith in the divine will. It comes from an alignment with the divine will and an identification of oneself as an instrument of the divine will. In this state, one can clearly see that which is not serving the higher purpose and remaining still within oneself, one can fight to quell falsehood. 
While passive equanimity prepares the inner stillness, active equanimity is the capacity to act from that stillness. Both of these approaches have their place and the two are not incompatible. 
Equanimity in The Parts of Being
Pain and desire are the two experiences of the physical being that can be dealt with equanimously. Pain or illness must not be met with revelry nor pity in one’s condition, but with a calm sense of detachment and faith. Desire in the physical being, such as desire for food can become a hindrance in the path of yoga. One must eat what is needed, when it is needed and not meet food with eagerness, nor repulsion.  
Likes and dislikes, desires and aversions arise from the vital being. Detachment from the movements of the vital being is an essential step on the path of yoga. Detachment does not mean forceful suppression of these movements but an equanimous observation of these movements as they pass through. In this way, we can learn to channel our attention towards the light that is waiting to be received instead of responding to the lower impulses of the vital. 
Cultivating equanimity in the mental being has been described as a delicate and difficult process, but a necessary one. Only in a state of mental equanimity can higher knowledge descend. The three reactions of the mind, ignorance, error and true knowledge must all be met with equanimity. One must be able to acknowledge one’s own ignorance, errors of the past as well as the true knowledge that one has gained without becoming attached to any of them. Equanimity in the mental being is the capacity to detach from one’s mental formations and allow new ideas to come in. 
The psychic being is the part of the self that is eternal. It is able to perceive all things as an expression of the Infinite and is able to treat all things, pleasant or unpleasant with a perfect equanimity. Once equality of the soul has been established, all other parts of being also begin to imbibe equanimity. On the other hand, the growing equanimity in other parts of the being prepares the individual for higher spiritual equanimity to descend. 
Why is Equanimity Important?
For Building Resilience
Equanimity can be seen as a safety net for an individual on the path of yoga. Without an equanimous attitude, the individual is vulnerable to all the disturbing forces in the environment and one’s own lower nature. One’s outer conditions may not always be conducive to inner spiritual progress. Having an equanimous attitude enables individuals to survive outer disturbances without becoming disturbed themselves. Inner disturbances such as feelings of illness and depression can also be dealt with if one is able to remain calm and not energise the feelings of illness or depression by worrying about them too much.  
For Overcoming Duality
One must be able to face both pleasant and unpleasant events in life with composure. This is the first sign that the individual is ready to rise above dualities and make progress on the path of yoga. Even the tiniest reaction of the ego is like a pebble thrown into a still pond, and can make ripples that stir the whole system. Equanimity in terms of ideas that are contradictory to our own also bring a deeper tolerance and a capacity to see that divinity is in all beings, only the expression varies. This also facilitates a movement from the ego which is bogged down by duality towards a space of oneness and with it comes the possibility to move towards a higher consciousness.  
For Moving Towards Higher Consciousness
Moving towards a higher consciousness requires a certain degree of emulation of that which is the highest; the divinity that looks upon all manifestations of itself with equality. This state of equality is the foundation upon which light (knowledge), force (power) or ananda (bliss) can descend. As this state becomes more intense, it begins to embody contentment and ease in all situations. Nothing can disturb it and it is not dependent on anything external. In this state, it becomes possible to experience universal love and a sense of oneness with all beings.  
How to Cultivate Equanimity
Conditions in Which Equanimity Can Manifest
Mental and vital quietude are essential conditions for equanimity to manifest. This is not a silence that comes from inertia but a silence cultivated by concentration. This is not the kind of silence that is complacent but the kind that makes one even stronger and resistant to attacks. Even in the most challenging situations, one must be able to sustain inner quietness; that will allow for one’s inner guidance to be heard. 
One must be in a state of detachment towards the fruit of their labour, to their work, towards praise and blame in order to cultivate a state of equanimity. If one is attached to their work or desirous of rewards, there can never be true equanimity. This detachment can also develop into a detachment from one’s own emotions, where it becomes possible to observe one’s anger, sadness or elation from a state of calmness and not get carried by the tide. 
Being in a state of oneness, the realisation that there the One is in all of us and we are all One, makes it possible for us to have a vast and empathic experience of all the different kinds of people and ideas that we come across. 
Steps Towards Equanimity
Corresponding to the above pre-requisites of quietness, detachment and oneness are the three steps towards becoming equanimous; endurance, indifference and submission, respectively.
The first step is to replace jugupsā,or contraction of the being in challenging situations with titiksa or the capacity to face all challenges. At first, we must learn to bear the blows that life brings us. This makes it possible to move to the next step that is a calm indifference to all happenings. 
Indifference is that capacity to move past attraction and repulsion towards situations, people and ideas that we come across in our life; having cultivated the confidence that we can endure them. It subdues the importance of the mind, the vital and the emotional being that become attached to their beliefs and puts the true self, the psychic being at the centre; which is detached and unmoved by the movements of daily life. Indifference does not mean to turn away from the happenings of life. It can be imagined as a high-seated presence that is not stirred by the movements of life.  
The third step is a surrender to the Divine Will and an acknowledgement of the self as an instrument of the Divine Will. This enables us to act from a place of equanimity where it is necessary to act. It paves the way for an active equanimity to manifest. 
Read more about Equanimity from the works of the Mother and Sri Aurobindo.