In quantum mechanics when we talk about the wave nature of particles are we referring in fact to the wave function? Does the wave function describes the probability of finding a particle (ex: photons) at some location? So do the "waves" describe probabilities just the way in classical physics the electromagnetic waves describe the perturbations of the electric and magnetic fields?
No, because the wavefunctions are not waves in space. They are waves in enormous high-dimensional spaces of possibilities. If you have two particles, the wavefunction is waving in 6 dimensions (the two positions of the two particles make a six dimensional space of possibilities), if you have three particles, the wavefunction is in 9 dimensions. So it is always wrong to think of it as a wave in space, like a field.
There is a field which obeys the Schrodinger equation, but this classical field is a classical wave, like E and B, which describes many coherent bosons in the same quantum state all moving together, like a superfluid or a Bose-Einstein condensate.
The key point to understand about wave/particle duality is that when we describe some system (e.g. an electron) as a wave what we mean is that it interacts like a wave. Similarly when we describe it as a particle we mean it interacts like a particle. The electron itself is neither a wave or a particle: it's, well, an electron.
The other point is that we can describe our system using various mathematical approaches. When you say wavefunction I'd guess you're thinking about the solutions to the Schrödinger equation. The Schrödinger equation is basically a wave equation so it works very well when describing wave-like interactions. It can be used to describe particle-like interactions, but this gets messy because you have to model your particle as the superposition of infinitely many waves.
You're quite correct that the wavefunction describes the probability of finding the particle, but the wavefunction is not simply a wave like a sine wave.
Usual quantum mechanics is roughly based on following principles : 1. For any given ("small") physical system $S$ there is associated a set $H_S$ of physical states. 2. At any instant of time $t$ system $S$ exists in some state $a_t\in H_S$. Time evolution of this state is governed by a first order (in time) differential equation called Schrodinger equation. 3. State $a_t$ carries all information about the system that one can hope to get. This information is probabilistic and depends upon what 'observable' you want to measure. In particular if you want to measure position you will see a particle, if you want to measure momentum you will see a wave. However words could sometimes be misleading; so you should consult some good text e.g. Cohen, Tannoudji Volume 1.
Edit : "Wave" as this term is used in QM does not mean ordinary physical wave but mathematical "probability wave". So when we talk about wave nature of particle what we are referring to is the position space probability wave (or wave function ) associated with it. So you are right :-)
Does the wave nature of a particle refer to the wave function?
No, and for a very simple reason. The wave nature of a quantum particle refers to the empirical evidence - observations - that quantum particles, like the electron and the photon, exhibit classical wave like properties such as interference and diffraction.
The wave function of a quantum particle is part of a mathematical model that, in the non-relativistic limit, accurately predicts both the wave like and particle like observations. The wave function's magnitude squared is interpreted as a probability density.