First of all, as has been stated in this thread before, de Broglie's ideas are heuristic. It was a crucial step in the history of the discovery of quantum theory, and it's worth being studied today in the very beginning of studying quantum theory. In fact, I don't know any other way to make quantum theory comprehensible than this historical heuristic arguments, although in general, I don't like the historical approach to physics, because the strength and aim of the natural sciences lies in the possibility to formulate simple fundamental principles, which allow to derive observable predictions from mathematical analysis. In the case of quantum theory, however, the most important thing is to develop a physical understanding of this mathematical formalism, and for that the historical approach is valuable, because one understands, how the modern form of quantum theory (first formulated by Dirac in 1925 and then mathematically sharpened by von Neumann).
De Broglie's idea was to write down a wave equation for particles, motivated by Einstein's idea of "wave-particle duality" of electromagnetic waves (light). It should be stressed at that point that this idea is nowadays known to be inaccurate and that in modern quantum theory there is no such concept. De Broglie's idea thus was that also for "material particles" like electrons there are some "wave" or "field" aspects to describe them in terms of quantum theory.
What he couldn't know is, that his heuristic only works for the non-relativistic case, and that's why I like to discuss this case only here. De Broglie tried to do this for relativistic particles, but that's totally flawed and shouldn't be taught anymore today, because the (only hitherto known) right way to formulate relativistic quantum theory is quantum field theory, and for that you need heuristics which can be explained only when you have the complete quantum-theoretical machinery for many-body quantum physics at hand.
From the wave-particle duality picture it's clear that a free particle with a sharp momentum and energy is described by a plane wave,
$$\psi(t,x)=A \exp(-\mathrm{i} \omega t+\mathrm{i} k x),$$
where ##\omega## is the frequency and ##k## is the wave number of the wave. Now, to derive an equation of motion for these waves you need a dispersion relation, i.e., the function ##\omega=\omega(k)##. This is well-known in classical field theory. There are different dispersion relations for different kinds of waves, derived from the underlying field equations of motion, e.g., from hydodynamics (sound waves in gases and liquids, water waves, etc.) or classical electrodynamics (electromagnetic waves, light), etc. Here we want to do the opposite: From the solutions we want to guess the equation of motion for the field.
Now comes the idea to use Einstein's "wave-particle duality" idea: The energy and momentum of the particle is related with the plane-wave parameters via the Einstein relations,
$$E=\hbar \omega, \quad p=\hbar k,$$
here written in modern form with the reduced Planck constant ##\hbar=h/(2 \pi)## to make the notation more convenient.
Now for a free non-relativistic particle, the relatition between energy and momentum is
$$E=\frac{p^2}{2m}.$$
This implies
##\hbar \omega=\frac{\hbar^2}{2m} k^2 \; \Rightarrow \; \omega=\frac{\hbar}{2m} k^2.##
Now, it's easy to find a linear equation of motion for the plane wave. To get one power of ##\omega## down you need a time derivative and for two powers of ##k## a second-order ##x## derivative. Indeed for the plane wave written down above we have
##\partial_t \psi(t,x)=-\mathrm{i} \omega \psi(t,x), \quad \partial_x^2 \psi(t,x)=-k^2 \psi(t,x).##
Now, glancing at the dispersion relation, we multiply the first equation with ##\mathrm{i} \hbar##, and the second equation with ##-\hbar/(2m)##, which gives
##\mathrm{i} \hbar \partial_t \psi(t,x)= -\frac{\hbar^2}{2m} \partial_x^2 \psi(t,x).##
This is nothing else than the time-dependent Schrödinger equation for free particles moving along the ##x##-axis, derived from de Broglie's argument, which earned him not only his PhD degree (after Einstein had given an enthusiastically positive evaluation of the thesis, which de Broglie's thesis advisors where unsecure about what the think of it) but also a Nobel prize.
The next step in the historical development was the debate about what the meaning of ##\psi## might be. Einstein, Schrödinger and some others thought it's something like a physical field, analogous to classical electromagnetic fields, and ##|\psi|^2## might be the densitity of particles. On the other hand, as the above "derivation" shows, ##\psi## described a single particle and not many. A classical-field interpretation thus would imply that an electron is some wide-spread "smeared" quantity, like an electromagnetic field. On the other hand, up to today there's no such thing as a "smeared electron" ever observed, but only single electrons making a pointlike spot on a photo plate. This lead to the today (by most physicists at least) accepted "interpretation" of the wave function, which is due to Born in the perhaps most important and famous footnote in his paper on quantum-mechanical scattering theory: ##|\psi|^2## is the probability density to find a particle at position ##x##.
This immediately implies that the plane waves, describing a free particle with sharp momentum and energy, is not among the physically interpretable solutions, but for these you must be able to normalize the wave function (by choosing an appropriate value for the so far totally ignored amplitude ##A##) such that
$$\int_{\mathbb{R}} \mathrm{d} x |\psi(t,x)|^2=1,$$
which says that the probability to find the particle somewhere on the ##x## axis is ##1##, i.e., that the electron for sure must be somewhere in space.
Now comes the important point that we have written down a linear field equation of motion, and that there are plenty of solutions that are normalizable in the above given way, which can be very easily constructed by using Fourier integrals. The Einstein-de Broglie dispersion relations hold true for each plane-wave mode, but many (even "almost all") solutions are given by "linear superposition",
$$\psi(t,x)=\int_{\mathbb{R}} \mathrm{d} k \frac{A(k)}{\sqrt{2 \pi}} \exp(-\mathrm{i} \omega(k) + \mathrm{i} k x), \quad \omega(k)=\frac{\hbar}{2m} k^2.$$
It's easy to see that this function obeys the Schrödinger equation and is normalized correctly, if
$$\int_{\mathbb{R}} \mathrm{d} k |A(k)|^2=1.$$
Thus for any square-integrable function ##A(k)## you get a squar-integrable solution ##\psi## of the free-particle Schrödinger equation, a "wave packet", describing the probability distribution ##|\psi(t,x)|^2## to find it at at position ##x## any instant of time ##t##.
. I see. So what do people mean when they say stuff like "wavelength of the electron in electron diffraction experiment (Davison-Germer for instance)" or the "wavelength of the electron in the electron microscope?". Is this some sort of average of its component waves?
