Brahmi sthiti, a compound word according to Madhavacharya, means steadiness of the mind or intellectual stability. It is the spiritual status that consists of the stable or firm mind in Samadhi in the waking state and grounded in the Absolute.
The term "Brahmi sthiti" appears at the close of the Second Chapter of the Bhagavad Gita in Sloka II.72 :-
The Second Chapter of the Bhagavad Gita is devoted to the Samkhya Yoga, a spiritual discipline, which is one of the six astika ('orthodox') schools of Hindu philosophy that was founded by Kapila. Samkhya elucidates the eternal Truth of the Self. In the context of the discipline of knowledge and that of action, the Samkhya Yoga teaches that the Self, being free from the six-fold transformation like birth and so on, is a non-agent. It is in this chapter that Krishna tells Arjuna that the man who, giving up all objects of desires, moves about seeking nothing, and rid of all sense of "mine" and "I", wins peace (Bhagavad Gita Sloka II.71). Such a person who is of stable wisdom having known Brahman attains peace or Nirvana which is cessation of all forms of transmigratory sufferings. This attainment of peace or Nirvana is attaining the status of Brahmi sthiti, the status of Brahman, attaining which none gets deluded (any more), and abiding in it, at least at the hour of death, one gains super-consciousness in Brahman (Bhagavad Gita Sloka II.72). Adi Shankara in his Bhashya explains that the status of Brahmi sthiti is won by remaining anchored in Brahman after renouncing all objects of desires in their totality and after renouncing all works.
Thus, Brahmi sthiti means establishment in Brahman or grounded in the Absolute. It is also known as the Seventh state of consciousness i.e. the Unity Consciousness or Brahmi chetana, in which state the universal, transcendental value of the object of perception is appreciated and the object is found to be nothing other than self-referral consciousness; when the experience and the object of experience, both brought to infinite value of consciousness, are perceived as unified on the level of consciousness. Therefore, it is also known as the third Siddhi (perfection) called Brahma-bhuyam.
Brahmi sthiti or spiritual situation is the way of the spiritual or godly life; it is liberation from material bondage leading to the kingdom of God. Atman-ananda i.e. happiness of the Atman, that pertains to Brahmi sthiti is called Shanti, which experience is gained when one abandons all desires and lives and acts free from longing, who has no "I" or "mine" i.e. who has extinguished his individual ego in the One and lives in that unity, and attains to great peace; such a person stands firmly in Brahman  when one gets totally purged of his ignorance and delusion attains the highest state of perfection – Self-realisation.
The firmness of the mind, essential for attaining Self-realisation, is achieved through control of thought waves and by developing one-pointed concentration. Prabhavananda explains that control of thought waves advocated by Patanjali in his Yoga Sutras means the unlearning of the false identification of the thought-waves with the ego-sense which process involves a complete transformation of character. The sum total of samskaras is character, which character can be transformed by gain of right knowledge and control of all thought-waves by means of practice and non-attachment (Yoga Sutra I.7 & 12). Then, one develops one-pointed concentration firstly on a single object and thereafter in which the consciousness contains no object. Patanjali states that those forms of concentration which result in extraordinary perceptions encourage perseverance of the mind (Yoga Sutra I.35). Reaching nirvichara samadhi (super-reflective/seedless) or Nirvikalpa Samadhi one gains knowledge which is filled with truth that goes beyond inference and scriptures and which Samadhi wipes out all other past impressions, the pure mind uplifts from the subtle to the Absolute.