What we observe in nature exists in several scales. From the distances of stars and galaxies and clusters of galaxies to the sizes of atoms and elementary particles.
Now we have to define "observe".
Observing in human size scale means what our ears hear, what our eyes see, what our hands feel, our nose smells , our mouth tastes. That was the first classification and the level of "proxy", i.e. intermediate between fact and our understanding and classification, which is biological. (the term proxy is widely used in climate researches)
A second level of observing comes when we use proxies, like meters, thermometers, telescopes and microscopes etc. which register on our biological proxies and we accumulate knowledge. At this level we can overcome the limits of the human scale and find and study the enormous scales of the galaxies and the tiny scales of the bacteria and microbes. A level of microns and millimeters. We observe waves in liquids with such size wavelengths
Visible light is of the order of Angstroms, $10^{-10}$ meters. As science progressed the idea of light being corpuscles ( Newton) became overcome by the observation of interference phenomena which definitely said "waves".
Then came the quantum revolution, the photoelectric effect (Particle), the double slit experiments( wave) that showed light had aspects of a corpuscle and aspects of a wave. We our now in a final level of use of proxy, called mathematics
The wave particle duality was understood in the theory of quantum mechanics. In this theory depending on the observation a particle will either react as a "particle" i.e. have a momentum and location defined , or as a wave, i.e. have a frequency/wavelength and geometry defining its presence BUT, and it is a huge but, this wavelength is not in the matter/energy itself that is defining the particle , but in the probability of finding that particle in a specific (x,y,z,t) location. If there is no experiment looking for the particle at specific locations its form is unknown and bounded by the Heisenberg Uncertainty Principle.
What is described with words in the last paragraph is rigorously set out in mathematical equations and it is not possible to understand really what is going on if one does not acquire the mathematical tools, as a native on a primitive island could not understand airplanes. Mathematics is the ultimate proxy for understanding quantum phenomena.
Now light is special in the sense that collectively it displays the wave properties macroscopically, and the specialness comes from the Maxwell Equations which work as well in both systems, the classical and the quantum mechanical, but this also needs mathematics to be comprehended.
So a visualization is misleading in the sense that the mathematical wave function coming from the quantum mechanical equations is like a "statistical" tool whose square gives us the probability of observing the particle at (x,y,z,t). Suppose that I have a statistical probability function for you, that you may be in New York on 17/10/2012 and probabilities spread all over the east coast of the US. Does that mean that you are nowhere? does that mean that you are everywhere? Equally with the photons and the elementary particles. It is just a mathematical probability coming out of the inherent quantum mechanical nature of the cosmos.
Light and matter are neither particles nor waves. We use the particle and wave analogies to allow us to apply some level of intuition to the effects. The interference effects in light (and matter interference experiments) follow similar equations to waves in water so when we want to talk about the interference, we call light and matter a wave. When we want to talk about effects such as the absorption of a discrete quanta of light in the photoelectric effect or the billiards like collisions of two atoms in a gas, we talk about light and matter as particles.
Reiterating: light and matter are neither particles nor waves... they are something else, something different than a sum of the two catch phrases. That difference generates amazing effects that defy our macro-world based intuition.
As for calculations, quantum field theory (QFT) treats both matter and light in a way that handles both the wave and particle like effects in one theory. One thing to note: light and matter have significantly different properties in QFT. The 'particle' properties of light are not the same as the 'particle' properties of matter.
Best Answer
That's a deep question. The idea of particle-wave duality is behind the theory of quantum mechanics. In quantum mechanics the state of a system, be it a free particle for example, is given in term of a wave-function $\psi$ given by Schrodinger's equation $$i\hbar\frac{\partial\psi}{\partial t} =\hat{H}\psi$$ Whenever you measure something on a state, it collapses on a given eigenvector and the result of that measurement is given in term of the eigenvalue corresponding to that eigenvector.
The simple example is the one of a free particle, be it a photon, in one dimension for which the hamiltonian is just $$\hat{H} = \frac{\hat{p}^2}{2m} $$ so that Schrodinger's equation reads
$$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}$$
for which a solution is the following wavefunction
$$\psi(x,t) = e^{\frac{i}{\hbar}\left(px-\frac{p^2}{2m}t\right)} $$
The only measurable quantity for a free particle is it's momentum. Without going into the details, if you measure the momentum from this wavefunction you'll get $p$. So, by doing an experiment, what you'll measure is a particle with momentum $p$.
The theory for photons is actually more complicated than this since you absolutely need a relativistic formulation of quantum mechanics, but the idea behind it is just about the same.