In modern physics, duality most often refers to the paradigm underlying quantum mechanics, according to which matter can exhibit properties associated with wave physics as well as classical particle mechanics.
Because these two sets of phenomena are mutually exclusive in classical physics, but nevertheless are both needed in order to describe the possible behaviors of matter, one speaks of the wave-particle duality.
This duality also implies that light and electromagnetic radiation, which in classical physics are considered to be wave phenomena, can exhibit properties of matter: light has to be described as a wave in order to explain interference effects, whereas the photoelectric effect is usually explained by invoking the particle nature of light (although it can be shown to be fully explicable by treating light as a wave as well).
Dual descriptions of a physical system may have overlapping ranges of validity. In such regions of overlap, two dual theories will not give contradictory predictions.