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.
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
Rox, I highly recommend that you get a copy of Richard Feynman's QED: The Strange Theory of Light and Matter. You are asking some interesting questions, but you will need to state them more precisely before you can get an answer that will be fully satisfying to you. QED is both one of the most interesting physics reads I know of on the oddness of things quantum, and simultaneously one of the most precise. Feynman wrote it for a non-mathematical friend, and avoided using any equations (well, except in some footnotes, just to brag about the truly amazing fit of complex numbers to the problem of quantum mechanics). Unlike many pieces on this subject, Feynman will not lead you astray with false or glitzy analogies. He realized that reality itself is quite, quite weird enough without any window dressing.