How can I explain the Quantum/Wave theory to my class?
You can't! The folks who have postulated the quantum nature of matter and wave-particle duality, and other quantum theories have trouble explaining it in terms other than mathematical equations. When trying to explain it in conceptual terms, we're asked to accept things that don't make sense, and in some ways, physicists use this as justification that the theory is correct.
Anyway, here are a couple ideas for discussing the quantum/wave properties of energy.
Until around 1900, when Max Planck developed the idea of quanta, energy had been thought to be a phenomenon of continuous flow - basically waves. Quantum theory describes energy as separate, discrete "particles". An analogy you could use to explain the wave/quantum idea might be that of analog versus digital. In the analog sense, energy flows in continuous streams or waves, having no specific inherent quantity - in other words, an energy wave could be any size. The quantum idea says energy is a "digital" flow, that what appears to be continuous waves is actually broken down into discrete, individual "bits". The name "photon" is used for these individual energy particles. Photons contain a specific amount of energy. For example, if you have a pure red light (like a laser), it can be thought of as a stream of photons all having a specific energy (the units for measuring this energy are usually electron-volts). The more photons, the brighter the light - but all the photons individually have the same amount of energy. In fact, these individual particles of energy can be detected discretely, or counted.
Now, at the same time, the light has the properties of a wave. The wave can be described by its wavelength and frequency. Experiments can be devised that show light (or other electromagnetic energy) to act as a wave or a particle. If the quantum theory is correct, we must accept this "weirdness of science" even if we can't explain it very well.
Unfortunately, it gets worse when we look at properties of matter. In the subatomic world, it was discovered that particles such as electrons, which are usually thought of as actual physical chunks of something, can be observed behaving as if they too are waves. It turns out that by using the equations that describe photons as waves, one could describe an electron as having a certain wavelength and frequency - matter waves.
You might use some demonstrations or "thought experiments" to attempt to show this difference. Here's an idea (you could almost try this, or something like it). Imagine a stretched string. On the string is a cup that can hold ping pong balls. The further you pull the string back, the more balls you can get into the cup. You pull the cup back and load one ball. You let go, and the string vibrates with a given wavelength and frequency - the sound you hear has a certain pitch. And the ball goes flying out with a certain specific momentum - a quanta of energy. If we keep the tension the same, we always get the same pitch and the ball has the same energy. If we tighten the string, the higher the pitch of the sound it makes when "plucked", and the more momentum the ball has (in some cases, we are only allowed to vary the tension in discrete steps). Now, we can only increase the loudness of the sound by pulling back further on the string. But we are only allowed to put in 1, 2, 3.. balls. So the "intensity" or amplitude of the sound increases in discrete steps (number of balls). This is true of electromagnetic energy - you can't have one and a half photons of light. The brightness increases by adding photons. And for any given pitch, the balls all have the same momentum (energy). Again, with light, a given color (pitch) has photons of a specific inherent energy. We can think of the sound as the wave version of the energy, and the balls as the particle version.
Of course, like any analogy, my example is not a rigorously correct one, but maybe it will give you ideas. And, we have to remember that a theory is only a model that seeks to describe how the physical world behaves. There are other theories which describe the same properties which are addressed by quantum theory. Apparently, some of these theories are also mathematically correct and contain easier to understand conceptual explanations that don't seem to conflict with reality. But they often require us to imagine ten or more dimensions.
Keith Welch, Radialogical Controls Group (Other answers by Keith Welch)