Another in my continuing series trying to explain some condensed matter concepts in comparatively jargon-free language. So far I've talked about electron-like quasiparticles, phonons, and plasmons. Now we consider magnons, also known as "spin waves". A magnon is another collective excitation, like a phonon or a plasmon, that may be described by a wavelength (or equivalently a wavevector) and an accompanying frequency. In phonons, we were interested in the pattern of atomic displacements away from their equilibrium positions, and we thought about this in a balls-and-springs picture of solids. Magnons, as the name suggests, are intimately related to magnetism. In many materials there are magnetic moments associated with (some or all of) the atoms in the material, and you can think of these moments as little arrows. In a material with "ferromagnetic interactions", the system can lower its energy by having the moments tend to align with each other. In a true ferromagnetic state all of the moments spontaneously align - all of the arrows point in the same direction. Flipping one arrow 180 degrees around would cost quite a bit of energy, since that arrow would then be antialigned with its neighbors. On the other hand, it costs much less energy to move one arrow just a little bit out of alignment with its neighbors. A magnon is a collective excitation where the relative alignment between neighboring magnetic moments is spatially described by some wavelength (That is, start at some arrow. Translating over by one magnon wavelength takes you back to an arrow tilted the same way as the initial arrow.).
Now, when you tilt a magnetic moment in a magnetic field, that moment will feel a torque that will cause it to precess. This is completely analogous to a tilted gyroscope precessing when it feels a gravitational torque. So, each little moment participating in the magnon is precessing around, giving a time-dependence to the local moment orientation.
This has been a very classical description. Quantum mechanics enters in a couple of ways when talking about real materials. First, there are quantum mechanical restrictions on what we can say about different components of an electron's magnetic moment at any one time. Second, like phonons, one can think of these magnons a bit like harmonic oscillators - a given magnon mode with angular frequency \omega can only exchange energy in chunks of size \hbar \omega.