Non-duality is a term is used to refer to a group of similar experiences that may be described in various ways. Sometimes a person may feel like he is a universal consciousness that consists of all beings and all things. This may be called an experience of oneness, or a unitive experience. Another person might perceive that all that exists is pure consciousness and the self is really an illusion produced when the mind discriminates between different things. This may be called an experience of no-self. Another person may further observe the interdependence of mind and matter and apprehend that there is no firm foundation for either. This is sometimes called an experience of emptiness.
The discussion below could be accepted by a materialist because it describes what people feel they experience subjectively, it doesn't require belief that consciousness is not produced by the brain. However, for completeness, I should point out that most people who have a deep experience of non-duality believe that they perceive that consciousness is not limited to the mind of one person but consists of all consciousness and all things in the universe. The discussion is also not exactly correct. Non-duality, reality undifferentiated by words and concepts, cannot be described with words and concepts. And different experiencers also describe it slightly differently and may disagree on the fine points of what the experience is.
Non-duality is the idea that if a mind does not distinguish between things, then everything can be considered one big thing. When one looks at the universe without distinguishing between different things, everything and everyone is just one big thing. Without a mind to think, "this is a chair", "this is a car", "this is a tree", "this is my self", "this is something other than my self", there is no duality, there is no multiplicity, there is just what is.
For example, suppose you have a room with a chair in it. "Chair" is a human word, a concept that does not exist outside a mind. Without a mind to look in the room and perceive a chair, there is no "chair" anywhere. There is just the undifferentiated non-dual reality. The matter may exist in its form, but "chair", a concept, does not exist until someone thinks "chair" or recognizes a chair. When a human looks into the room, the human sees what is in there and discerns there is a chair. The concept "chair" now exists in his mind. In that instant, duality is created by the mind, for the mind, in the mind.
A person who has realized non-duality, will understand this creation to be the illusion that hides the ultimate reality, the non-dual reality, from most people.
A person experiencing a non-dual mode of consciousness would not experience any separation between himself and the chair. He might, in an experience of oneness, feel that the chair is himself, or in an experience of no-self, feel the chair is the knower.
Once, when a Zen Buddhist teacher gave an introductory class in Zen, he showed a small bell to the students and asked, "What is this?" One student said, "It's a bell." The teacher was silent. Another student took the bell from the teacher and rang it for everyone to hear. The teacher said, "Very good!" The second student was demonstrating non-dual consciousness.
Consciousness cannot exist without anything to be conscious of. So a chair, giving the mind something to be conscious of, creates the mind. The mind creates "chair" and the chair creates the mind. Interdependent, without any firm foundation.
Self is just like that. The mind sees and feels its body and creates "self" just like it created chair. The mind perceives feelings, sights, sounds, tastes, smells, impulses, opinions, awareness, liking, disliking, wanting, not wanting, and creates "self" and "other" just like it created "chair".
In the undifferentiated reality, I and my friend are not different people and we are not the same person. Without mind to think "same" and "different" there is no same or different. There is just what is. Not two. Not one.
Non-duality, when realized due to direct experience and understanding, produces happiness, contentedness, compassion and selflessness. When you make no distinction between yourself and everything and everyone else, you don't worry about yourself, you don't want anything, and you have the same concern for others' suffering that you have for your own. However, the experience does not automatically make you a saint. It does not necessarily produce personality changes that will make everyone love and admire you. And there have been ethics scandals involving a number of spiritual teachers.
Samadhi, a non-dual experience that can be produced by meditation, is the experience of "the before thinking mind" i.e. non-duality, undiscriminated reality, reality undifferentiated by mind. Samadhi is the experience of reality before the mind creates a multiplicity of things and self.
Without the experience of samadhi, the understanding of non-duality is superficial. Without the understanding of non-duality, the experience of samadhi is meaningless. Together they can produce a realization of non-duality.