Since OP is clearly asking about the so-called wave-particle duality of Quantum Mechanics, I'll narrow my answer to that.
In classical mechanics, particles are point-like objects in space and time, while waves are distributed phenomenon spread out through space and time and are solutions to wave equations. These are models.
In QM, the words particle and wave are used very loosely, and all things considered, are terrible choices for describing what is being modeled in QM because they carry too much suggestive baggage from CM. In QM, the things being studied are "wavefunctions" (unfortunate name because of wave being in it), which encode everything there is to know about a quantum system, and the "measurements", that act upon wavefunctions to extract physical information about wavefunctions.
The way wavefunctions change in time when nothing is interacting with them satisfy the wave-like Schrodinger equation (this is why they are wave-like).
However something very special happens when you measure a wavefunction. For each type of measurement (position, momentum, etc) you can only get certain values, and each of these values is associated with a specific wavefunction. It turns out that any wavefunction can be written as a "blend" of these special wavefunctions, and when you measure a wavefunction, it turns exactly into one of the special wavefunctions (associated with the value you just measured) that is part of the blend. This is probabilistic, with special wavefunctions that contribute more being more likely to be measured.
That's pretty abstract, so let's now explain why people talk about "wave-particle duality" crap. In empty space, when you try to measure the position of a wavefunction, the special wavefunctions are point-like (to the accuracy of your measurement device). If you measure a wavefunction and get (say) x = 1m, the wavefunction "collapses" to a point at 1m. Afterwards, if left undisturbed, it evolves like a wave per Schodinger's equation.