The Paradox of Waves and Particles
Let's take the archetype of waves, sea-waves. What are they, waves or particles? Well, they are waves made of particles (water molecules). So if you have a volley ball floating on the surface of the sea, its behavior will be the behavior of a body that interacts with a wave. It is hit by millions of molecules of water, the behavior of which can only be determined on a statistical level, and described by wave-type theories.
However, if instead of a classical ball we consider a body of a size comparable to the size of water molecules (e.g. a molecule of carbon that floats in the sea), its behavior will be the behavior of a body that interacts with particles (the molecules of water). It will collide with individual molecules, and in this case we can determine its behavior by particle-type theories.
So it seems that most if not all waves are waves and particles at the same time, or to be exact, are waves formed by particles. The particle-like "quantum" of a sea wave is a water molecule that moves up and down and that, together with its neighboring molecules that perform the same movement with a phase difference, constitute the wave. If a body of comparable size (the carbon molecule) encounters the "sea wave quantum", it will be knocked off its course by a particle-like collision, even though it interacts with a wave.
So, is a sea wave a wave or a particle? The question becomes rather pointless. It is a wave formed by particles, and when the interaction is on a large scale it manifests as a wave, while if it is on a scale comparable to the size of the particles that form the wave, the existence of the particles becomes evident. Nothing mysterious or paradoxical here.
Of course, this does not address other aspects of the wave-particle duality, such as what happens when elementary particles behave like waves. A first attempt to demystify this matter is the Single Slit Experiment Simulation.
And finally, this obviously leaves us with the big question: what are the "particles" that form the electromagnetic waves? You will find some initial attempts at an answer in About h.