The wave function ψ is related to probability since P (x ∈ (0, x1)): = ∫ |ψ|² dx; x ∈ (0, x1), where P (x ∈ I) is interpreted as the probability of finding the particle in the interval I.
note that the variation of this function |ψ|² with the x-axis, indicating the most likely places.
note that in trying to solve the Schrödinger's EDP we get a family of solutions that depend on an n; n ∈ N.
This phenomenon is interpreted as a quantization of energy, in the sense of having discrete amounts of energy.
Therefore, the differences in ψ_n are related to different distributions of probability (I am not entering into the merit of who implies who).
ψ for physicists is related to something that they call "probability amplitude", this ψ function, does not assume only real values, but also assumes complex values.
what it means? is a reason for discussion in many books... but the main point is its relation to probability (described above).
for probability we can speak of many analogues, but for the amplitude of probability, I can't!
you can think of ψ as a mathematical device to generate the "right" probability distribution function.