What is a wave? From sound and water waves we come to an association with sine and cosine variational behavior. Wave equations are differential equations whose elementary solutions are sinusoidal .
In water waves and sound waves and even electromagnetic waves what is "waving", i.e. has a sinusoidal variation with time and space, is the energy of the wave, represented by its amplitude.
When dimensions become very small, compatible with h, the Planck constant the individual "particles" electrons etc., can be described sometimes like classical billiard balls, and at the same time they exhibit a randomness, which when accumulated displays interferences and other wave characteristics.
This single electron at a time double slit experiment shows both effects. The individual electrons leave a point on the screen which seems random. The accumulation gives a probability distribution that has sinusoidal variations.
One can only give a probability for the electron to appear at the (x,y) of the screen, which depends on the quantum mechanical solution of the boundary value problem "electron scattering from two slits"
So it is not a classical particle behavior because even though the energy is carried by the single electron, its (x,y) is controlled by a probability distribution; and it is not the classical wave, i.e. a single electron that is "waving" its mass all over the screen interference pattern. Each electron is a quantum mechanical entity.