For light, wave-particle duality works like this. To understand some aspects of how light behaves, such as interference and diffraction, you treat light as a wave. To explain other aspects (photoelectric effect and Compton effect) you treat light as being made up of particles. Light exhibits wave-particle duality, because sometimes it acts like a wave and sometimes it acts as if it is made up of particles.
Things we usually think of as particles exhibit wave-particle duality too. The behavior of relatively large objects, like baseballs, is dominated by their particle nature; to explain the behavior of very small things like electrons, however, both wave properties and particle properties have to be considered.
Examples of particles showing wave properties: