What contributions did Schrödinger make to atom theory?
He devised an equation that allowed (when solved under the constraint of the particular situation described) to find a WAVEFUNCTION, representative of the basic characteristics of the system studied.
Schrödinger equation is very "complicate" looking and in its "simpler" version (one dimensional and no time) looks like:
In essence it allows you to find the Wavefunction that represents the PROBABILITY of finding an electron (for example) in an atom that has a certain "shape" (given by the potential energy term) and at a certain energy.
The wavefunctions look like the red curves (after some ...small...manipulations!) and give you a map of probability distributions, the dotted areas (where it is more probable to find your electron):