Advaita is a profound philosophy.  It is the philosophy of knowledge and understanding.  In the history of philosophies advaita emerged much later, probably after the naiyaka, vaisheshika and Sankya philosophies.  Even though they were advaita gurus in ancient times, it was Sri Sankaracharya who made it available to the layman. 

In advaita philosophy, understanding is more important than any spiritual practice.  Advaita philosophy is a two step teaching and understanding.  The first is the discrimination between the Atma and anatma.  It is otherwise known as viveka.  At present we assume ourselves to be a body with a conscious mind, and some are not even aware of the conscious mind.  They simply assume themselves to be a body and that is why when the body dies they imagine that everything is lost.  According to advaita philosophy, each individual is a conscious being. 

Let us understand this with a traditional example.  Few pots filled with water are kept in the garden.  The quality and quantity of the water may differ.  Sun light falls on all the pots.  The reflection of the sunlight and the image of the sun may differ according to the quality of the water.  If the water is pure then the reflection and image are good.  The reflected sun is only an image and so it may not have the qualities of the original sun like the heat, brightness and radiation.  Likewise, the body is compared with the different pots.  The water is compared with the different minds and the sun is compared with the supreme consciousness.  That one supreme consciousness is reflected on different minds.  Only mind has the property to reflect the Consciousness.  The reflected consciousness is called chitabasa.  Every mind has a conscious ego “I”  It joins with other thoughts and it becomes our personality.  An individual or jiva is a combination of body, mind, and the reflected consciousness.  This combination is anatma; Anatma is inert.  It cannot perform its action without the pure supreme consciousness.  Atma, which is the supreme consciousness is intelligent.  Hence Atma is different from anatma.  This is the first step of understanding.

The second step of understanding is, I am that supreme consciousness.  I am that Atma which is all pervading.  It is the ultimate understanding.  Anatma is the reflection of the Atma.  The mind has different impressions and all the impressions become alive because of the Atma; and advaita says Atma is Satya (existence) and anatma is only reflection (maya).  Advaita philosophy never denies the universe; it only says that it is the appearance (mitya) and not the reality.  The individuals are appearances or reflection and the reality is Atma.  When the individuality is removed from all of them what remains is Atma.  Brahma satya jagad mitya: it is the core of Advaita philosophy.

These are the two steps in understanding advaita philosophy.