Turiya refers simply to the Fourth. There are four quarters to Brahman; the waking, dream, dreamless, and the Turiya. The Fourth state is not separate from the other three but rather transcends and encompasses the other three states as well as a Fourth. Turiya is Pure Consciousness and is the same as the Atman or the Unconditioned Brahman. The Mandukya Upanishad verse VII says:
नान्तःप्रज्ञं न बहिष्प्रज्ञं नोभयतःप्रज्ञं न प्रज्ञानघनं
न प्रज्ञं नाप्रज्ञम् । अदृष्टमव्यवहार्यमग्राह्यमलक्षणं
शान्तं शिवमद्वैतं चतुर्थं मन्यन्ते स आत्मा स विज्ञेयः ॥
(Swami Nikhilananda translation):
Turiya is not that which is conscious of the inner (subjective) world, nor that which is conscious of the outer (objective) world, nor that which is conscious of both, nor that which is a mass of consciousness. It is not simple consciousness nor is It unconsciousness. It is unperceived, unrelated, incomprehensible, uninferable, unthinkable, and indescribable. The essence of the Consciousness manifesting as the self [in the three states], It is the cessation of all phenomena; It is all peace, all bliss, and non-dual. This is what is known as the Fourth (Turiya). This is Atman, and this has to be realized.
The Upanishad then goes on to explain the meaning of AUM and the correspondence of the individual letters to the first three states and finally that AUM, in total, corresponds to Turiya. Verse XII then says:
The Fourth (Turiya) is without parts and without relationship; It is the cessation of all phenomena; It is all good and non-dual. This AUM is verily Atman. He who knows this merges his self in Atman--yea, he who knows this.
Sankara's and Gaudapada's commentary on this verse then explain that those who meditate on AUM in the heart as a symbol of Brahman will realize AUM, will realize Brahman.
In his introduction to his translation of the Mandukya Upanishad and Gaudapada's Karika, Swami Nikhilananda gives some general comments on how to realize Brahman. He says (The Upanishads, V2, p 206):
Therefore rigorous disciplines are indispensable for the actual experience of Reality. The Upanishads ask the pupil to practice various disciplines, such as austerity (tapas), continence (brahmacharya), truthfulness (satyam), and reverence (sraddha). Vedanta lays down four general disciplines. These are discrimination between the Real and the unreal, renunciation of the unreal, a group of six virtues (calmness of mind, control of the senses, self-settledness, forbearance, concentration, and faith), and lastly an irrepressible longing for the Knowledge of Reality.
The last discipline (longing, or as Sankara says in may places, a deep hankering) is considered by many the most important. A person who has this hankering will naturally develop the others. Ramakrishna Paramahamsa gave the following story: A student kept on asking his teacher what was needed to realize God. One day, the teacher had the student go down to the lake with him. After they entered the water, the teacher grabbed the student's head and pushed him underwater and held him there for sometime. After a while the teacher let go of his head and let him surface. The teacher asked the student - "What were you thinking?" the student replied - "I could think of nothing else but wanting a breath of air!" The teacher replied - "When you long for Brahman as you did for that breath of air, Brahman will be realized."
When you can think of nothing else but Brahman to the exclusion of all else, the Turiya will be realized.