Self-realization or Inner self is a concept that has become widely popular in the Western World. It has been greatly influenced by some Eastern religions. For instance, for the Hindu religion self-realization refers to a profound spiritual awakening where there is an awakening from an illusory self identify image (Ego), to the true, divine, perfect condition that the individual is. The branch of Advaita Vedanta is the one that has especially developed this concept.
Furthermore, the method of meditation Sahaja Yoga, created in 1970 by Shri Mataji Nirmala Devi, defines self realization as a connection with your self or the first encounter with reality.
One of the definitions in the Western can be found in Merriam Webster's dictionary. It defines self-realization as “fulfillment by oneself of the possibilities of one's character or personality”.
Also, Abraham Maslow and Carl Rogers, American psychologists, developed the concept of self-actualization in Humanistic Psychology. Maslow defined then self-realization as “the impulse to convert oneself into what one is capable of being.” 
Based on Maslow, the most common meaning given to self-realization is that of psychological growth and maturation. It represents the awakening and manifestation of latent potentialities of the human being -for example, ethical, esthetic, and religious experiences and activities.
Aajit K. Das, in the International Journal for the Advancement of Counselling, compared and contrasted Maslow and Rogers' concept of self actualization with the concept of self-realization in Vedandic Hinduism and the two major schools of Buddhism: Theravada and Mahayana. The author concluded in this paper that the two concepts complement each other.