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.
Well, I would say the electromagnetic field is the medium.
I don't understand this - How can something be the medium for itself?
Light isn't just an electromagnetic field. Light (i.e. photons) is a wave---a disturbance or perturbation---in an electromagnetic field. Just like for gravity, fields apparently permeate space. Even massive particles (e.g. quarks or electrons) can be described (or viewed) as waves in underlying fields.
So... maybe photons make up light, and they are just shooting through space like a wave, but it really isn't a physical wave (like a sound or water wave), it's just "acting like one"?
What you're calling a 'physical wave' is an emergent property of a underlying medium (e.g. air, water). For the most part, such an emergent wave is basically the same as a wave in a field. 'Physical waves' are often referred to as 'quasiparticles', because of this similarity. What we think of as 'particles' (e.g. electrons) don't just 'behave like waves', they are also waves, hence the 'wave-particle duality'.
Best Answer
I'd like to add a slightly different take on both Anna V's and Ben Crowell's answers.
Classical optics IS the theory of one photon. There is NO approximation in this statement in free space (classical optics is actually a little bit more than this generally, but I'll get to that below). Maxwell's equations to the lone photon are EXACTLY what the Dirac equation is to the lone electron (indeed the two equation sets can be written in forms where they are the same aside from a mass term in the Dirac equation coupling the left and right circularly polarized fields together, whereas the two polarisations stay uncoupled in the mass-free Maxwell equations). Photons are Bosons, which means you can put as many of them as you like into the right same state: so you can build up classical states which correspond EXACTLY to one photon states. When you do experiments where one photon is transferred at a time, you solve Maxwell's equations for a field inside your experimental set up, normalise the solution field so that the total electromagnetic field energy is unity, and then the energy density $\frac{1}{2}\epsilon_0 |\mathbf{E}|^2 + \frac{1}{2}\mu_0 |\mathbf{H}|^2$ becomes the probability density that your one photon will be photodetected at the point in question.
The full quantised theory of light works like this: each monochromatic free space mode (plane wave) is replaced by a quantum mechanical harmonic oscillator (which you have likely dealt with). The motivation for this is that free space classical plane waves oscillate sinusoidally with time and so a classical free space wave is assumed to corresponds to a coherent quantum state of the corresponding quantum harmonic oscillator. You may recall that energy may be given to/withdrawn from a quantum harmonic oscillator only in discrete packets. It is these discrete packets which are the "particles". In this bigger picture, a one photon state is a quantum superposition of next-to-ground-state (one photon) states of the infinite collection of plane wave harmonic oscillators that are "THE FIELD"; the spatial Fourier transform of the superposition coefficients propagates precisely following Maxwell's equations. Otherwise put in the Heisenberg picture: in a one photon state, the electric and magnetic field observables evolve with time precisely following Maxwell's equations. Note that in this description, it's very hard to say where the "particles" actually are: this is how I like to think of it: "the Electromagnetic Field communicates with the outside world (the other electron, quark, ... fields making up the universe) in discrete data packets, and these packets are what we call photons". Also note that, in the light of Anna V's comments below: one doesn't have to stick with plane waves: one can choose any complete orthonormal set of fields and assign quantum harmonic oscillators to each of these and the description is wholly equivalent. So one chooses whatever the basis states make the analysis of their particular problem easiest.
Going back to our one photon state evolving following Maxwell's equations: when we put dielectrics and other matter into the description, we no longer purely have photons. If we represent the "atoms" of the matter by two level quantum systems, when light propagates in matter it is not really just light: it is a quantum superposition of free photons and excited matter states. The description of lossy materials in this picture is a little more complicated but it can be done: lossy materials are continuums of quantum oscillators that the photon has an extremely low probability of remission from once it's absorbed there). So here then is how I like to think of classical optics:
Classical Optics = The Theory of One Photon + Optical Materials Science
Let's go back to the statement about putting many Bosons into the same state and thus building a classical light field that is mathematically the same as a one photon state. Mostly that's all there is to macroscopic optics: and this is what I believe Dirac meant when he said famously that "each photon interferes only with itself". In macroscopic states built by simply copying Bosons, it should be pretty clear that whether you calculate their propagations wholly separately as lone photons and then sum up their probability densities to get the field intensity, or if you simply classically calculate the field intensity, you'll get the right same result. Most macroscopic light fields behave like this and it is actually very hard to find deviations from this behavior. Aside from the experiment where we turn the light level right down low so that interference patterns are built up "click click click" one photon at time, the photon is extremely hard to observe experimentally as a quantum and not as a classical light field. Mathematically what all this means is that macroscopic states behave as though they are products of one photon states (special Glauber "coherent" states - actually discovered by Schrödinger): and the last twist in the quantum optics tale is the phenomenon of entanglement. This is what we see in the seldom and very hard to set up situations where the "product" behaviour no longer holds: but you may care to see the Wikipedia page on quantum entanglement or ask another question to find out about that one!
Lastly, to think about laser trapping, as Ben says indeed a classical field theory is enough. What you might be thinking of is laser cooling and in this case Anna V's comments apply. Here the atom is withdrawing one quantum at a time from the electromagnetic field and likewise emitting a few quantums at a time. But it's not all "particle-like" - for instance, the Fermi golden rule calculations that give the cross sections for these particle transitions all involve overlap integrals between the atomic dipoles and the wave fields - note that this is a very classical looking calculation wholly analogous to the analysis of the interaction between a short ($\ll \lambda$) dipole antenna and classical electromagnetic field. As in Ben's answer, both the wave and particle aspects of the photon's behaviour are thus showing themselves here. Also - I might be guessing here as laser cooling is not my field - an optical photon's momentum is quite an appreciable chunk of a slow atom's momentum. Optical photons are of the order of 1eV so that their momentum is of the order $10^{-27}\mathrm{kg\,m\,s^{-1}}$ and a proton's mass is of the order of $10^{-27}\mathrm{kg}$, so the transfers are way too chunky to be reduced to continuous momentum / energy flux calculations and still hope for an accurate picture.
Some further reading on the subject of quantum optics can be found in R. Loudon, "The Quantum Theory of Light" and the first chapter of Scully and Zubairy, "Quantum Optics".