How does the Wave-Mechanical model reconcile with the particle definition of an electron?
Wave particle duality is a fundamental idea in the quantum mechanics required to understand the behaviour of fundamental particles

One of the key ideas in quantum mechanics is that particles fundamental behaviour cannot be explained if we assume their "real" behaviour is Newtonian (or billiard-ball like). The behaviour of particles like electrons depends on how we measure them and sometimes they appear wave-like sometimes particle like.

This is very important in chemistry as it leads to good explanations of chemical bonding and the shapes of molecules. Electrons in atoms and molecules, for example, were originally thought to behave like a mini solar system where they orbited the nucleus. But this explanation doesn't work well to explain much in chemical bonding or atomic behaviour.

Schrödinger developed a better way to think about their behaviour with his famous wave equation. This describes the shapes of electron clouds around atoms. One way to think of the output of this equation is that it describes clouds of probability describing the possible locations of an electron in an atomic orbital. Those orbitals are the "standing waves" of electrons bound to atoms and they have specific definite shapes (related to spherical harmonics which are the 3D equivalent of the standing waves in a vibrating violin string). 

This description of the properties of electrons is the core of our understanding of chemical bonding and the shapes of molecules. 

What are electrons? Quantum mechanics doesn't tell us what they really are: it just tells us how they behave when we make certain measurements. And, in chemistry, the measurements that matter mostly give results where the wave-like behaviour is what matters.