Schroedinger equation:
(-ħ²/(2m))∂²ψ(x,t)/∂x² + U(x)ψ(x,t)  = iħ∂ψ(x,t)/∂t.
The Shroedinger equation tells us how the wave function evolves.

Interpretation:
We interpret |ψ(x,t)|² as a probability density.  The probability of finding the particle at time t in an interval ∆x about the position x is proportional to |ψ(x,t)|²∆x.

How do we predict the outcome of a measurement?
We have to "operate" on the wave function.  Every measurable quantity is associate with an operator, something we have to do to the wave function.