Amazingly, the stages of sleep that the Upanishad talks about are quite similar to what modern science shows us. Except that the latter derives no higher goal from sleep. As we saw in the Mundaka Upanishad, such a knowledge about the material world is only lower knowledge. (Note: The Upanishads do not despise a lower knowledge) Just as the Mundaka suggests that there is a higher knowledge which is Self Realization, the Mandukya also strives to know about the Self, and the mode of this realization is through sleep.
As per the Upanishad, consciousness is in four stages, awakeness, dreaming, sleep, deep sleep and an even deeper super conscious state. It associates the sounds of the three letters 'AUM' to the first three stages, and the complete "AUM" to the fourth stage. The following is the English translation of the Upanishad.
1) AUM stands for the Supreme Reality.
It is a symbol for what was, what is,
And what shall be. AUM represents also
What lies beyond past, present and future.
2) Brahman is all, and the Self is Brahman.
This Self has four states of consciousness.
3) The first is called Vaishvanara, in which
One lives with all the senses turned outward,
Aware only of the external world.
4) Taijasa is the name of the second,
The dreaming state in which, with the senses
Turned inward, one enacts the impressions
Of past deeds and present desires.
5) The third state is called Prajna, of deep sleep,
In which one neither dreams nor desires.
There is no mind in Prajna, there is no
Separateness; but the sleeper is not
Conscious of this. Let him become conscious
In Prajna and it will open the door
To the state of abiding joy.
6) Prajna, all - powerful and all - knowing,
Dwells in the hearts of all as the ruler.
Prajna is the source and end of all.
7) The fourth is the superconscious state called
Turiya, neither inward nor outward,
Beyond the senses and the intellect,
In which there is none other than the Lord.
He is the supreme goal of life. He is
Infinite peace and love. Realize him!
8) Turiya is represented by AUM.
Though indivisible, it has three sounds.
9) A stands for Vaishvanara. Those who know this,
Through mastery of the senses, obtain
The fruit of their desires and attain greatness.
10) U indicates Taijasa. Those who know this,
By mastering even their dreams, become
Established in wisdom. In their family
Everyone leads the spiritual life.
11) M corresponds to Prajna. Those who know this,
By stilling the mind, find their true stature
And inspire everyone around to grow.
12) The mantram AUM stands for the supreme state
Of turiya, without parts, beyond birth
And death, symbol of everlasting joy.
Those who know AUM as the Self become the Self;
Truly they become the Self.
References & further reading:
 The Upanishads - Eaknath Easwaran
 Mandukya Upanishad - translated by Vidyavachaspati V Panoli
 Mandukya Upanishad - wikisource link
 Mandukya Upanishad - Hindu Website link
 Mandukya Upanishad - with Gaupada Karika