Wave behavior of particles When people say that every moving particle has an associated wave, do they mean that the particles will move up and down physically, for example when we say that a moving electron has a wave associated with it, does the electron physically oscillate?  Or is it some other wave, like a probability wave? I really don't understand the latter.
 A: I agree with Pieter, especially when he refers to QED, writing

Feynman explains this (in his little book QED, I recommend to read
  that) with electrons having dials that turn around. Or one could
  represent phase as color.

It must be remarked however that Feynman was a strong opponent of the
so-called "wave particle duality. Let me quote from p. 15:

I want to emphasize that light comes in this form - particles. It is
  very important to know that light behaves as particles, especially for
  those of you wo have gone to school, where you were probably told
  something about light behaving as waves. I'm telling you the way it
  does behave - like particles.

And in note 3 at p. 23:

No reasonable model could explain this fact, so there was a period
  for a while in which you had to be clever: You had to know which
  experiment you were analyzing in order to tell it light was waves or
  particles. This state of confusion was called the "wave-particle
  duality" of light, and it was jokingly said that light was waves on
  Mondays, Wednesdays, ans Fridays, it was particles on Tuesdays,
  Thursday, and Saturdays, and on Sundays, we think about it! It is the
  purpose of these lectures to tell you how this puzzle was finally
  "resolved".

I entirely agree with Feynman. His purpose was to show how the
wave-like behaviour of particles could be modeled by assigning to each
particle an "amplitude" - a complex number, which Feynman depicts as
an arrow. 
In some notes of mine I further elaborated this point. Here is a
translation (original is in italian):

Conviction about wave character of light was based on a series of
  experimental facts [...] we only recall by their names:
  interference and diffraction. It was known, and it is easy still
  today to verify in the laboratory, that all known waves give rise to
  interference and diffraction: this holds true for surface waves on a
  liquid as well as for sound waves. From here it was concluded: since
  waves exhibit interference and diffraction and light too exhibits
  interference and diffraction, it follows that light is made of waves.
It is easy to see the on logical grounds such conclusion is
  unwarranted: all fishes live in water, dolphins live in water, then
  dolphins are fishes? We don't mean that for a century physicists had
  fallen into a so trivial logical error, but simply that it was an
  induction, not a deduction. It was very plausible, on the ground
  of the analogy with wavelike phenomena, to induce that light too were
  made of waves. Experimental sciences proceed this way very often, and
  are often successful. But it may also happen that ensuing research
  shows the induction was unjustified. This is what happened with light.
Our reasoning cannot stop at this point however, because if we
  assert light is made of particles (photons) we still have to explain
  how it can behave in ways we are used to attribute to waves. At least
  we must be prepared to accept that if we are dealing with particles,
  they are particles sui generis, very different from the idea of
  particle our everyday experience could suggest.

A: They do not mean that the particle itself moves up and down in a wavelike way. What is meant by wave/particle behavior or duality is something more subtle; something that many very smart people have spent their lifetimes working on and which my answer will treat in a simplified way which I hope you can grasp. 
For objects that are very, very small, it is possible for them to manifest radically different behavior depending on the details of their environment, whether they are singly isolated or part of a large population, and how they interact with the tools we use to detect and study them. 
For example, in the case of a single electron that is speeding through space, it's possible to interrupt its path with a detector that registers the impact of the electron as if it were a tiny bullet. It is also possible to interrupt a stream of electrons with a detector that bends their paths just as if they were a train of waves instead of a stream of tiny bullets. 
The standard interpretation of the so-called wave equation that describes the propagation of an electron through space is that the probability of finding the electron at a given location along its path can be extracted from that equation, and that this probability varies from point to point in a way that is wavelike: the crests of the wave represent spots where the electron is most likely to be found, and the troughs represent spots where it is least likely to be found. 
There is a strict and well-defined mathematical formalism which is used by skilled practitioners to handle questions like this, and there are others here who can furnish you with that if you wish.  
A: The wave is there to describe the phenomena of diffraction and of interference. Particle beams can interfere destructively: no intensity at some spot when both beams are on.
This can be described by a phase and the mathematics of waves. When phases are opposite, the sum is zero. Feynman explains this (in his little book QED, I recommend to read that) with electrons having dials that turn around. Or one could represent phase as color. 
But physically, there is no transverse wave oscillating up and down. Physically, there is no dial. Physically there is no color. These are just representations of phase, which mathematically describes the phenomena of many different kinds of waves.
A: In addition to Niels answer, I want to add a few details.

... the probability of finding the electron at a given location along its path can be extracted from that equation, and that this probability varies from point to point in a way that is wavelike: the crests of the wave represent spots where the electron is most likely to be found, and the troughs represent spots where it is least likely to be found.

If one shots a beam of electrons to a detector, the exposured spot could be have a Gaussian distribution or something similar to this. The spot will be bright in the centre and of vanishing intensity away from the center. All depends from how to focuse the beam. No trace of a wave-like distribution.
On the over hand, since H. Boersch in 1940 published his paper
"FRESNELSCHE Elektronenbeugung" (Wikipedia is available only in German language, but some information I have collected here, especially on page 7) it is known that behind edges electrons behave somehow similar to light. Both get bended under the influence of an edge. The difference is, that light is bended also behind the geometrical shadow, while electrons are bends only away from the edge and it’s geometrical shadow. But in both cases the edge influence the particles in a way they get distributed with swelling intensity behind the edge (the so called fringes).
These fringes were measured (I hope so) and a sine function of this distribution was claimed. This has allowed to find a wave equation for this distribution.
All else ist abstraction and imagination. One could conclude that - since the distribution looks like a wave - the beam of itself is a wave. Or one could conclude that the interaction between the beam and the edge forms the intensity distribution on the observers screen. But the last is a speculation of mine.
A: We don't really know. The ontology of physics at the quantum level is a bit of a mixed bag. Consider that the Lagrangian of the Standard Model has over a hundred terms. When people describe physical law as simple they surely do not mean this! 
The nearest ontology to what you're asking for is Bohmian Mechanics though I think it was actually originated by de Broglie who actually came up with the suggestion that physical particles like electrons could also be waves. Here, though, the particles are guided by a pilot wave. 
A: The wave of an electron really does oscillate in time. Every wave function has two components (mathematically, we treat this using a real and imaginary part of the wave function). These two components rotate into each other sinusoidally. One component might have 100% probability and the other zero, and then they will switch. Think of one component as being the sine and the other the cosine. However, when you measure the electron, the probability you see is only the squared sum of the two components, sin^2 + cos^2 =1, and so you wouldn't see these oscillations.
In reality, however, a wave function does not have a fixed frequency. Instead, it is a wave packet with a Fourier decomposition containing many frequencies. These waves do interfere with each other, leading to the broadening of the wave packet with time (the electron spreads out). But you would not see the oscillations themselves unless you do something very clever.
