Duality means that everything that exists is double, always in pair with another, never alone. Everything - word, thought and act - has it's partner without which it does not exist. This partner is it's exact opposite. It's not possible to have one without the other just as it is not possible to have one side of the coin without the other.
For example, the word give. It's opposite is: take. One does not exist without the other. Giving cannot exist without taking. If there is someone who is giving something, there must be someone who is taking it, for who would be able to give anything if there was no one to receive it? This truth applies to all that is in the universe. The universe is filled with pairs of opposites: black - white, good-evil, up-down, left-right, wise-stupid, etc. These are just a few examples of the infinite list of pairs of opposites, which we experience constantly although we often have no idea how they influence ourselves.
A pair of opposites constitute the essence of our lives. If we experience joy, then sooner or later, we will have to experience sorrow; if we are happy for some reason, then we will be unhappy for another; if there is a beginning, there also must be an end; night follows day; and after the storm comes peace, etc. These pairs of opposites are woven into our lives, and we have to experience them whether we want it or not. It seems that our fate is already doomed, but it's not. We just need to find a way out of this vicious circle of cause and effect, action and reaction, caused by the limitations of our mind.
The way out exits, and it leads to the world of oneness, which means non-dual reality, where there are no opposites. According to our ancient Advaita-Vedanta non-duality philosophy, there is just One Spirit in the Universe, and everything, living or inanimate, is an inseparable and indivisible part of this One Spirit. Non-duality further says that it is only an illusion, caused by the mind and play of the senses, which makes us regard the world and everything in it as real and separate from us.
Imagine a state in which you feel oneness with the Universe, enjoying bliss and peace of mind, and at the same time acting and functioning normally in your day to day life. A state of being active in the world, yet maintaining a state of inner detachment. In this state you are aware of your oneness with the One Spirit, and also aware that the One Spirit is acting and manifesting through everything; things, plants, trees, animals and people. This is the experience of nonduality in it's highest form.
Ordinarily, the veil of thoughts and the five senses draw the mind outside, to the external world, and obscure the awareness of the consciousness that is beyond the mind. Meditation brings peace to the mind and develops the ability to silence it, thus enabling us to experience the "state" of non-duality. In this state of inner silence, one rises above the illusion of identification with the mind, thoughts and ego, gets beyond the illusion of separateness and realises oneness with the One Spirit. It is as if a new sort of consciousness dawns, and one sees the world in a different way.