Vedanta is the philosophy that developed out of Vedic oral traditions and scriptures. It is translated as the 'conclusion' of the Vedas, which includes texts like the Upanishads, the Brahma-sutras and the Bhagavad-Gita. These writings are even older than the Bible and there are scholars who see the origin of all major religions in these 'revelations of Truth'. The Jnana Yogi Ramana Maharshi once said that the whole Vedanta is contained in the two Biblical statements: "I am that I AM" and "Be still and know that I am God."
According to Vedanta philosophy, the human soul is part of All-awareness (Brahman). The human soul got involved with mental activity and is reborn until this circle is broken through remembrance of clear Awareness or Consciousness. It compares with a dream: We are reborn into the dream-state again and again unless we can find a way to stay awake. As all scripture and sages teach, waking up from our mortal dream is indeed possible.
Shankara, in accordance with the philosophy of Vedanta, teaches that there is only one Reality, one God (Brahman). Everything else is subjected to change and is no more than an expression of Reality. When seen as Reality, the world is an illusion (a notion that is now becoming popular with quantum physicists).
Since most folks prefer a simpler belief system, Shankara taught his philosophy mainly to Brahmins and intellectuals, who had no problems understanding Brahman as the omnipresent, eternal, Consciousness. They understood this Awareness as the essential Self within and everything else as a phenomenona mere appearance. His reasoning was resting on logic, the interpretation of the scriptures, but also on direct knowledge.
The philosophy of non-dualism is called Advaita and Shankara's philosophy is sometimes called Atmadvaita.