Advaita Vedanta is a type of Hinduism. People who believe in Advaita believe that their soul is never really different from God. The Sanskrit word for soul is Atman and the Sanskrit word for God is Brahman. The most famous Hindu philosopher who taught about Advaita Vedanta was Adi Shankara who lived in India more than a thousand years ago. Adi Sankara learned the sacred texts of Hinduism, like Vedas and Upanishads under his teacher Govinda Bhagavadpada and later wrote extensive commentaries of Hindu sacred texts called Upanishads. In these commentaries, he proposed the theory of Advaita, saying that the Upanishad actually teach that the individual soul (called Atman) is not different from God (called Brahman). He also taught that there is only one essential principle called Brahman (means God) and everything else is a kind of expression of that one Brahman. Because of this theory of one being, his teachings became popular as the "Advaita" (a = not, dvaita = two, means no-two or non-dual). The way he said this to people was "Atman is Brahman."
Adi Shankara was very smart and knew that people would wonder how he could say such an odd thing. He realized that many people would ask him, "If a person's soul is really one with God all along, then what makes a person feel so separate from God?" His answer to this was that we are ignorant of our real self being God because we see through a kind of filter—like looking through a dirty piece of glass—and he called this filter we look through, maya, which means "illusion" in Sanskrit. Shankara said that our ignorance makes us feel very separate from God, and even from everything around us. Shankara suggested that the best way people can find the truth is for them to try to clear their thinking of all ignorant thoughts, be very good, and think very hard about who they really are. He said that if a person did all these things he would realize that God was himself all along.
This is a very different idea from some other religions where we are told we are separate from God and need to go to Him. In Advaita, the idea is that people never really were separate from God, but their ignorance made them see it that way.
Not all Hindus believe in Advaita Vedanta. Many others believe there is a difference between a person's soul and God, just like in many other religions.