Electrons - What is Waving? If an electron is a wave, what is waving? 
So many answers on the internet say "the probability that a particle will be at a particular location"... so... the electron is a physical manifestation of probability? That doesn't sound right. 
This page http://mwolff.tripod.com/see.html seems to suggest that the spherical wave pair is what gives the electron the properties of a particle. I could be misinterpreting, but this answer makes more sense to me - if only I understood what was waving!
 A: According to quantum electrodynamics (QED), which encodes the properties of electrons and photons, electrons are excitations of an electron fiueld in the same way as photons are excitations of the electromagnetic field. 
The fields wave, and the electrons (or photons), as far as they can be considered to be particles, are localized wave packets of excitations of these fields. 
The particle interpretation is appropriate, however, only to the extent that the so-called geometrical optics approximation is valid. This means, in a particle interpretation, you shouldn't look too closely at the details, as then the particle properties become more and more fuzy and the wave properties become more and more pronounced.
But if you just look at quantum mechanics (QM) rather than QED, your question cannot be answered as the wave function is something unobservable, existing only in an abstract space, 
What can be given an interpretation in QM are certain things one can compute from the wave function. The stuff of interest to chemists is the charge distribution, given by $\rho(x)=e|\psi(x)|^2$ for a single electron, by 
$\rho(x)=\int_{R^3} ~dy~ e|\psi(x,y)|^2$ for a 2-electron system, etc.; here $e$ is the electron charge. For electrons in a molecule, nothing is waving here anymore, as the wave aspect is eliminated by taking the absolute values. 
Indeed, if this charge density is concentrated in a tiny region, one sees the particle aspect of electrons; if it is very spread out, one sees the wave aspect, revealed by high frequency oscillatory patterns in the charge density.
This is the chemist's interpretation. See Chapter A6:The structure of physical objects of 
A theoretical physics FAQ.
Physicists (especially if not well acquainted with the use of charge density information) are often brainwashed by the teaching tradition, and then think and express everything in terms of probabilities, giving QM an unnecessary flair of mystery.
A: In QM, the "waves" lie on the Hilbert space, that is not our "real life" space (where we live). All we can do with this "wave", or with this wave vector (that could be called $\psi(x)$ or $|\psi \rangle$, depending on your choice of representation), to make prediction about the properties of a particle (here on your real lifes), like a electron in your question, its to calculate its square modulus:
$|\psi(x)|^2 \mathrm{d}x  \equiv P(x, x+\mathrm{d}x) $,
where $P(x, x+\mathrm{d}x)$ is the probability to find the electron between $x$ and $x+\mathrm{d}x$, if you want to measure the position of the electron; the rule above its called the Born Rule.
Well, this said, it's time to go to your question: "If an electron is a wave, what is waving?".
First of all, what determines the wave or corpuscular behavior of your subject of study its the type of experiment that you do; but, the outcomes (the results of the measure process) will always be probabilistic. So, in QM, we "accumulate statistics", to make a histogram and compare with the prediction of the Born rule to the system in question. 
Therefore, the electron could be a wave (existence of interference of electrons) or a particle (existence of quantas), depending on what you want to know about it. Both behaviors are complementary, that is, with the two descriptions the properties of the electron can be achieved completely, but the two cannot be measured simultaneously.
A: In QM, a "wave" isn't what we normally imagine: something that moves up and down and moves in one direction, like water. It's just a function that evolves with time and has a (in general) different value at every point in space. See this applet for some examples of atomic orbitals which are infact electron wavefunctions (the applet actually shows the absolute value squared $|\psi|^2$ of the wavefunction;  or the probability density). The wave does not "exist" per se in physical space. It can be drawn (superimposed) on physical space, but that just means that it has a value at every point there.
The wave associated with an electron shows the probability of finding it at a particular point in space. If an electron is moving, it will have a "hump" in its vicinity, which shows it's probability at every point in time. This hump will move just like the electron does. For more info on this (though you may have read stuff like this before), see the "why don't they need to be close" section of this answer. When you observe the electron, you collapse the hump to a peak. This peak is still a wave, just narrowly confined so it looks like a particle.
Your issue is that you're trying to look at the "electron" and "wave" simultaneously. This isn't exactly possible. The wave is the particle. You can look at it as if you exploded the electron into millions of fragments and spread it out over the hump. There is a fraction of an electron at every point. The fraction corresponds to the probability of finding it there. At this point, there is no electron-particle. So there's nothing that's "waving".  Of course, we never see a fraction of an electron, so these fellows clump together the minute you try to make an observation.
Edit by OP -- This is the section referenced above that I found most helpful

Quantum mechanics has a nice concept called wave particle duality. Any
  particle can be expressed as a wave. In fact, both are equivalent.
  Exactly what sort of wave is this? Its a probability wave. By this, I
  mean that it tracks probabilities.
I'll give an example. Lets say you have a friend, A. Now at this
  moment, you don't know where A is. He could be at home or at work.
  Alternatively, he could be somewhere else, but with lesser
  probability. So, you draw a 3D graph. The x and y axes correspond to
  location (So you can draw a map on the x-y plane), and the z axis
  corresponds to probability. Your graph will be a smooth surface, that
  looks sort of like sand dunes in a desert. You'll have "humps" or
  dunes at A's home and at A's workplace, as there's the maximum
  probability that he's there. You could have smaller humps on other
  places he frequents. There will be tiny, but finite probabilities,
  that he's elsewhere (say, a different country). Now, lets say you call
  him and ask him where he is. He says that he's on his way home from
  work. So, your graph will be reconfigured, so that it has "ridges"
  along all the roads he will most probably take. Now, he calls you when
  he reaches home. Now, since you know exactly where he is, there will
  be a "peak" with probability 1 at his house (assuming his house is
  point-size, otherwise ther'll be a tall hump). Five minutes later, you
  decide to redraw the graph. Now you're almost certain that he's at
  home, but he may have gone out. He can't go far in 5 minutes, so you
  draw a hump centered at his house, with slopes outside. As time
  progresses, this hump will gradually flatten.
So what have I described here? It's a wavefunction, or the "wave"
  nature of a particle. The wavefunction can reconfigure and also
  "collapse" to a "peak", depending on what data you receive.
Now, everything has a wavefunction. You, me, a house, and particles.
  You and me have a very restricted wavefunction (due to tiny
  wavelength, but let's not go into that), and we rarely (read:never)
  have to take wave nature into account at normal scales. But, for
  particles, wave nature becomes an integral part of their behavior.
  --Manishearth Feb 14, 2012

A: To answer this question, we need to use a new model of the electron.
An electron can be modeled as a tiny charge that revolves at the speed of light in an orbit having the Compton wavelength as a circumference.
To validate this model, you can calculate the current produced by the revolving charge as it passes an observer near the orbit.  Using this current and the electron's area, you can calculate the magnetic moment, which is the Bohr magneton, identically.  
Also, the mass is trivial to calculate, as is the angular momentum.
Further, the mass is contained in the interior field within the orbital area, not in the charge, so the charge itself is not precluded from revolving at the speed of light.  The whole electron as a particle system, however, cannot be accelerated to the speed of light.
As to the question, "What is Waving", an observer near the orbit will experience an impulse each time the charge passes. The impulse from the charge spirals outward at the speed of light with a spacing of a Compton wavelength between succeeding impulses.  Hence, this spiraling impulse field of "Compton wavelets" are the entities that are "waving".
It is also to be noted that if you look at the spacings which the wavelets from two approaching electrons cross as the electrons move with respect to one another, it turns out to be the de Broglie wavelength, identically.
Actually, all of this is trivial to model, and so the reader is encouraged to try it.
Finally, we acknowledge that there will be interpretations of quantum mechanical phenomena versus the highly precise structure of this model that will be troubling to some.
