Newton held in the 17th century that light was composed of particles, and he developed a successful optical theory based on this assumption.
In the early 1800s, diffraction experiments by Young and Fresnel provided evidence for the wave nature of light: when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves. Even the wavelength of the light can be computed from the pattern. When Maxwell in the late 1800s explained light as the propagation of electromagnetic waves with the Maxwell equations, the view of light as wave was widely accepted.
However, in 1905 Albert Einstein explained the photoelectric effect by postulating photons, quanta of light energy with particle-like qualities. In the photoelectric effect, a metal plate is struck by light and emits electrons; the energy of those electrons is determined by the light's frequency, while the number of the electrons is determined by the light's intensity. This effect cannot easily be explained if light is assumed to be a wave. Einstein postulated that the frequency ν of light can be related to the energy E of its photons by
In 1924, Louis de Broglie claimed that all matter has a wave-like nature and related wavelength λ and momentum p by his equation
De Broglie's formula was confirmed three years later by guiding a beam of electrons (which have mass) through a crystalline grid and observing the predicted interference patterns. Similar experiments have since been conducted with protons and even with whole molecules, and the formula has been confirmed in every case.
The Planck constant h is extremely small and that explains why we don't perceive a wave-like quality of everyday objects: their wavelengths are exceedingly small. The fact that matter can have very short wavelengths is exploited in electron microscopy.
In quantum mechanics, the wave-particle duality is explained as follows: every system and particle is described by wave functions which encode the probability distributions of all measurable variables. The position of the particle is one such variable. Before an observation is made the position of the particle is described in terms of probability waves which can interfere with each other.
An intriguingly simple experiment, the double-slit experiment, summarizes the duality: Shoot electrons (or anything else for that matter) at a screen with two slits and record their position of impact at a detector behind the screen. You will observe an interference pattern just like the one produced by diffraction of a light or water wave at two slits. This pattern will even appear if you slow down the electron source so that only one electron per second comes through. "Classically speaking", every electron either travels through the first or through the second slit. So we should be able to produce the same interference pattern if we ran the experiment twice as long, closing slit number one for the first half, then closing slit number two for the second half. But no: the pattern won't emerge. Furthermore, if we build little detectors around the slits in order to determine which path a particular electron takes, then this very measurement will destroy the interference pattern as well.
The pattern is a result of the electron's wavefunction being diffracted by both slits and interfering with itself. The wavefunction is a complex valued function of space and time. The square of the magnitude of this function describes the probability of finding the electron at a given location at a given time. Interference is due to the fact that the square of the magnitude of the sum of two complex number may be different from the sum of the squares of their magnitudes.
The experiment also illustrates an interesting feature of quantum mechanics. Until an observation is made the position of a particle is described in terms of probability waves, but after the particle is observed, it is described as a fixed value. How to conceptualize the process of measurement is one of the great unresolved questions of quantum mechanics. The standard interpretation is the Copenhagen interpretation which leads to interesting thought experiments such as Schrödinger's cat. Another interpretation is the many-worlds interpretation.