What are matter waves made of and what is their speed?

E-M waves are not "made" of anything other than the current state of various fields. Matter waves are similarly not "made" of anything; they simply represent the wavelike behavior of things we treat as solid mass, e.g. electrons. The whole point of wave-particle duality is that you cannot separate one from the other.

We don't really look at 'matter waves' as... erm... matter waves anymore. The understanding of the concept has moved on considerably since de Broglie (but textbooks tend to lag behind newer ways of thinking) One physicist is alleged to have quipped, a propos 'matter waves': "nothing oscillates there".

Instead it's 'safer' to look at 'matter waves' in the following way.

Suppose we have a subatomic particle. In accordance with Schrödinger's equation a wave function $Psi(mathbf{r})$ is associated with the particle.

One of Quantum Mechanics' postulates (possibly the most important one) is that the wave equation contains all the information about the particle there is to know. These observables, like momentum $p$, are obtained by applying quantum operators (e.g. $hat{p}$) to the wave function.

One of the most important observables is the probability function $P$, which according to the Born interpretation is given by:

$$P(x,Delta x)=int_x^{x+Delta x}Psi^*Psi text{d}x=int_x^{x+Delta x}|Psi|^2 text{d}x$$

This is for the probability of finding a particle in a $text{1D}$ domain in a $Delta x$ interval, located at position $x$.

The probability density function is given by:

$$P(x)=|Psi(x)|^2$$

Below are some probability densities for some quantum states of a particle in a $1D$ box with infinite potential on the boundaries:

So what is really tangibly 'wavy' here are these probability densities $P(x)$. Rather than 'matter waves', think of them as probability waves.

These probability waves (and not so much the actual wave functions) explain the interference patterns in two-slit electron beam experiments that were responsible for the emergence of the matter/wave duality worldview and QM itself in the early 20th Century.

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There's also the minor issue of the unit of measurement of the wave function. For example for the aforementioned particle in a $text{1D}$ box the wave functions are given by:

$$Psi_n(x)=sqrt{frac{2}{L}}sinfrac{npi x}{L}$$ For $n=1,2,3,...$

The unit of measurement is:

$$mathbf{[Psi]}=mathbf{m}^{-1/2}$$

There's no mass in sight, as $mathbf{m}$ here stands for 'meter'.