What is the wave in an electron?
It's not a stupid question. In fact, Quantum Field Theory is the field of physics that seeks to answer exactly this question. In QFT, in addition to the electromagnetic field, there is a single electron field that extends throughout the universe. Stable ripples in the electron field constitute individual electrons. Every fundamental particle has a universe-wide field associated with it. There are quark fields, Higgs fields (which gives rise to the Higgs boson), photon fields (also known as the electromagnetic field), and so on. The larger the amplitude of a ripple at a certain location in space, the larger the probability of finding a particle there.
The electromagnetic wave is a classical theory while matter waves are quantum mechanical. The wave aspect is a mathematical abstraction which allows us to predict future quantum states of the electron with a known probability.
From the famous Double-slit experiment, it is clear that electrons do behave as wave as well as particle. When it is detected by geiger counter, "click" sound appears & no matter how greatly the voltage is decreased along the cathode tube, "click" & never "half click" appears. So, electrons always arrive at lumps like bullets. However, unlike bullets the probability of detecting electron at the backstop in front of the slits is not like bullet but like interference of waves like water waves. So, electron does behave as wave.
Waves of what? Waves of probability. The quantity that varies with wave like electric field in electromagnetic wave is $\Psi(x,y,z,t) = \psi(x,y,z)e^{-(iE/\hbar)t}$, a complex entity called wavefunction. The wave associated with the electron is purely mathematical construct. It doesn't describe the space-time variation of any measurable quantity. The wave rather relates to the probabilities of observing the electron at different space locations as a function of time.
Photons do have wavefunction but it is not the classical EM waves. It needs relativistic approach & is too subtle. However, it can be expressed by means of electric & magnetic field i.e. $\psi(x) = \begin{pmatrix} \vec{E} \\ ic\vec{B} \end{pmatrix}$. You can check this paper for more info on this.