The concept of a wavefunction is integral to quantum mechanics, the most successful model of reality Humans have yet devised. And still, no one agrees on what a wavefunction is, even though it correctly predicts the outcome of any experiment we throw at it.
The basic idea is that if you square the wavefunction, you will get a “probability density” that tells you how likely the particle will be in a certain place. The wavefunction itself evolves according to Schrodinger’s Equation. All of chemistry is essentially keeping track of complicated wavefunctions of atoms and molecules:
One of the crazy properties of the wave function is that measurement can collapse it, or at least, appear to do so. Some belive this is due to the fact that any measurement must disturb and change the system. Others argue for the interpretation that the wavefunction is just a representation of OUR knowledge of the Universe, so making measurements that increase our information will affect the wavefunction.
In a paper on arxiv, the arguments for each side of the “ontic” (wavefunctions have a reality independent of observers) vs. “epistemic” (wavefunctions represent someone knowledge of the Universe) debate are presented. Since no one agrees even on the basic interpretation of quantum mechanics, or the way to look at thought experiments like Schrodinger’s Cat (and its extensions, such as Wigner’s Friend), the one certainty is that that controversy is far from over. And of course, more predicted quantum weirdness becomes reality all the time. Like creating images using entangled photons that never interacted with the object they are imaging.