Duality is the relationship between two entities that are claimed to be fundamentally equally important or legitimate as features of the underlying object.
The precise definition of a "duality" depends on the context. For example, in string theory, a duality relates two seemingly inequivalent descriptions of a physical system whose physical consequences, when studied absolutely exactly, are absolutely identical.
The wave-particle duality (or dualism) isn't far from this "extreme" form of duality. It indeed says that the objects such as photons (and electromagnetic waves composed of them) and electrons exhibit both wave and particle properties and they are equally natural, possible, and important.
In fact, we may say that there are two equivalent descriptions of particles – in the position basis and the momentum basis. The former corresponds to the particle paradigm, the latter corresponds to the wave paradigm because waves with well-defined wavelengths are represented by simple objects.
It's certainly not true that Young was wrong and Newton was right. Up to the 20th century, it seemed obvious that Young was more right than Newton because light indisputably exhibits wave properties, as seen in Young's experiments and interference and diffraction phenomena in general. The same wave phenomena apply to electrons that are also behaving as waves in many contexts.
In fact, the state-of-the-art "theory of almost everything" is called quantum field theory and it's based on fields as fundamental objects while particles are just their quantized excitations. A field may have waves on it and quantum mechanics just says that for a fixed frequency $f$, the energy carried in the wave must be a multiple of $E=hf$. The integer counting the multiple is interpreted as the number of particles but the objects are more fundamentally waves.
One may also adopt a perspective or description in which particles look more elementary and the wave phenomena are just a secondary property of them.
None of these two approaches is wrong; none of them is "qualitatively more accurate" than the other. They're really equally valid and equally legitimate – and mathematically equivalent, when described correctly – which is why the word "duality" or "complementarity" is so appropriate.
The demonstration shown in the answer of another respondent, with the time frames showing how the interference patern builds up over time, is one of the best pieces of evidence we have about the wave particle duality of matter at the quantum scale. An intersting aspect in all these mysteries of nature, that I would like to express my opinion about, is the following:
Let us talk about photons, because they are the most missunderstood objects in quantum mechanics discussions.
Wave or particle?
Photons are particles every day of the week, not some days they are waves and some other days they are particles. They are as much particles as the electrons are. We know that from the distinct clicks we hear in our detectors when sufficiently low intensity light arrives at them. The wave property of the photon, or any other particle, is the wave function, and I assume we are familiar with the interpretation given to it, as the probality to observe the photon (or any other particle) at some position $x$ at some time $t$. That is to say that there is no way to tell were actually the photon is before we observe it. The wave function in the mean time occupies the whole of the space that is available for the photon to be in. It is important to undestand that photons of the same colour are all identical (they have the same energy).
Two slit experiment: Now let us see what happens when a photon approches the two slits. The wave function that represents the photon will pass through the slits like waves do. It will split into two waves and recombine to interfere on the aray of detectors on the other side. The maxima corespond to high probability, the minima to zero probability. The consequence of this is that the photon is most like to show up in one of these maxima and will only hit one detector, but we don't know which one. Likewise, we don't know which slit it has gone through. An interesting point to make here is this, there is no way that one photon will hit two detectors at a time. Any attempt, or trick we might do to determine which slit the photon has gone through, destroys the interference pattern as all wave properties are removed!
Conclussion: The interference pattern people had seen in the Young experiment when they did it, they observed the pattern forming instantly because they used high intensity light. But we discover the reality when we use very low intensity light. It is like you turn down the water tap, and you start getting droplets instead of that continuous flow you had when the tap was fully open. And we know that if we look closer we will see molecules of water.
For a deep discusion on all these, try to google: Richard Feynman's lectures at the university of Auckland, New Zealand, First Lecture. Very entertaining too! Try this link: http://vega.org.uk/video/subseries/8
Best Answer
Yes. Gravitational waves have been observed, and assuming that quantum mechanics is the right way to think about the universe, then weak gravitational waves of the sort that can be observed at LIGO can be thought of as coherent ensembles of 'graviton' particles.
Now, theoretically this picture is 'OK' because although any quantum field theory that describes gravity is nonrenormalizable, the energy scale at which we expect new physics associated with gravity to be detectable is extremely large ($\sim 10^{19} \text{GeV}$). Hence, we can use an effective theory valid at low energies to describe gravity in terms of particles, even if the 'true' theory of gravity valid at arbitrary energies or length scales is somewhat different conceptually.
Unfortunately, it's impossible with current technology to observe or manipulate single gravitons in a lab, so gravitons are still theoretical.