According to Advaita, only the innermost part of you is aware or conscious. The name in Sanskrit for this awareness is Atman. It's the part of you that's really you.
Now here's where it gets interesting. According to Advaita, your atman is the same as the underlying absolute reality of the whole universe, which is called Brahman. Brahman is the source of everything; it's what the universe really is. The classic analogy is that of a clay pot where the universe is equated to the pot and the brahman is the clay. The clay is not inside the pot and the pot is not exactly just clay. The clay pot is clay in the form of a pot. The clay pot is in essence just a manifestation of clay.
What distinguishes Advaita from other interpretations of the Upanishads is this: Advaita asserts that since there is only one Brahman, there is only one Atman. There's only one "me" and we are all it. We're all one "thing" - Brahman; and there is only one awareness - Brahman. And this one pure awareness is represented by the symbol of OM.
You have to examine the idea that only the atman (which is also the Brahman) is aware. This idea is more subtle than it seems at first glance. If you close your eyes for a moment and try to focus your attention on your inner self, you will be immediately aware of feelings, hopes, thoughts, desires, fears, and a general sense of yourself. This is the inner you, right? Well, no. According to Advaita, if you are aware of something, it isn't really you. The real you (the atman) is the part that's aware. It's not anything of which you are aware.
This kind of examination is called Viveka (discrimination) in Sanskrit. It is a main component of the traditional method of Jñana Yoga. If you keep doing it, you will discover that everything you currently regard as yourself (including your ego and mind) is not aware. The awareness in you is different from those things. No matter how hard you try, you can't focus your attention on the part of you that is aware. If you could, it would become something of which you are aware.
The implications of this theory get really interesting...