I see this issue being discussed very often on the 'net. Other than questions about relativity and the constancy of the speed of light, this issue of wave-particle duality is the most talked about in many public forums. I suppose that is highly understandable since this is one of the most puzzling aspect of what we have observed of our world. Unfortunately, I think there's quite a number of misconception regarding this issue of "duality", almost as much as the misconception about the Heisenberg Uncertainy Principle.
To be able to discussion this wave-particle duality, we must first examine what we mean by "wave", "particle" and "duality". Only after we fully understand what it is that we are talking about can we then search for what the combination of these words mean. Inevitably, when we talk about wave and particle, we are applying our classical understanding of what they are. A wave is simply an "oscillation" of something. We are familiar with water waves, etc. These waves then to "spread out", and most of us who have seen demonstration using ripple tanks, understands what waves are, and how they behave. Often, the phenomena of diffraction, interferences, etc. are attributed to properties of waves.
Particles, on the other hand, usually means an object that has a definite size and boundary in real, physical space. A ping-pong ball is a sphere with some radius. Beyond that radius, there's no more ping-pong ball. So the ball has a definite boundary in real space. We tend to define particles that way, i.e. they are discrete objects with definite size and shape that have finite extent in space.
So if you look at the definition and our understanding of waves and particles, we can clearly see that an entity cannot be both. The wave definition is simply incompatible with the particle definition, and vice versa. This brings us to the issue of duality. When we observe the behavior of light, electrons, neutrons, protons, buckyballs, etc. exhibiting both wave-like and particle-like behavior, we then use the term "duality" to describe such puzzling observation. Why? Because we have to "switch gears" when we talk about wave and particles. We use one type of description/formalism when we talk about the wave-like picture, and then we change our description/formalism to talk about the particle-like picture. Thus, the duality.
But here's something that most people who haven't studied physics are not aware of. In quantum mechanics, there is no such switching of gears! There is only one consistent description of the behavior of light, electron, protons, neutrons, etc.. QM doesn't have to drop one formalism and adopt another when light behaves as "particles" and then exhibit wave-like properties. All of these observations are consistently described using one, single formalism! And usually, when we have such an ability, we do not consider this as a "duality". So in QM, there is no such thing as a wave-particle duality!
I'm sure this might come as a surprise, because the phrase "wave-particle duality" came about mainly due to QM and in the context of quantum behavior. However, one only needs to satisfy oneself that this duality doesn't exist by simply browsing through any undergraduate QM text. This duality is even hardly mentioned since it is utterly irrelevant. You certainly do not see such a thing being an issue in physics research papers other than papers that deal with pedagogical issues of quantum mechanincs.
So if there's no such thing as duality of wave and particles in QM, why is it still being used, especially when physicists talk about such a thing in the context of popular science? For better or for worse, physicists have to convey things to the public in ways that the public can understand! What a wave is, and what a particle is, are both well-known concepts. So we tend to talk about "wave-particle duality" to convey the message that these quantum objects can mimic those behaviors. However, most tend to skip the fact that these seemingly incompatible behavior can come out of just consistent formalism without the need to switch to a different one.
As I've mentioned before, QM has to be understood starting with the mathematical formalism. It is the ONLY thing that connects it to the rest of what we have already understood.