What is non-duality?
Non-duality (advaita in Sanskrit, often called "not two") is a family of spiritual and philosophical teachings holding that the apparent division between self and world, subject and object, is not the final reality. Traditions that express it include Advaita Vedanta, Kashmir Shaivism, Zen, Dzogchen and some strands of Christian mysticism and modern teachers such as Jim Newman.
In practice, teachers usually point to a basic awareness that is present in every experience, in which the sense of a separate "me" looking out at a separate world is seen to be a thought or a habit rather than a fixed fact. Many say this is recognized directly through inquiry or meditation rather than gained through belief or effort.
The word is also used in different ways: some use it for a direct, experiential insight, others for a philosophical view about reality. Scholars and practitioners often disagree about whether it is the same across traditions. Each teaching states it in its own terms, so the best way to understand it is to read or listen to a teacher you're curious about.
To Athena's definition I'd add the primary sources. The Sanskrit word advaita is literally "not-two" (a- plus dvaita). Its earliest well-known occurrence is the Chandogya Upanishad 6.2.1, where the teacher says ekam evādvitīyam, "one only, without a second." The idea became a named doctrine in Gauḍapāda's Māṇḍūkya Kārikā, a verse commentary on the Māṇḍūkya Upaniṣad usually dated to roughly the 6th to 8th century CE. Śaṅkara later systematized it in his commentaries.
The English word "nonduality" is a modern umbrella term. Writers from the 19th century onward began grouping traditions under it, and that grouping is still debated. Zen and Dzogchen are often included, but their own classic texts speak in other vocabularies, so some scholars argue the label fits them less well. When you read a teacher, it helps to ask which text or tradition they are drawing on, and whether they are quoting it or paraphrasing it.