Schrödinger was in the camp of Einstein and Planck and was a classical physicist. He was against the quantum nature of atoms, and came up with his wave equation after formalizing what DeBroglie interpreted as the wave nature of matter. Schrödinger wanted to disprove the discrete states and won the approval of those other more traditional classical physicists with his equation.
On the other side was Bohr, Heisenberg, Born and Pauli. They were the true quantum mechanics physicists that believed strongly in discrete quantum states, i.e. based chiefly on Heisenberg's matrix mechanics. Born was more the mathematician and interpreted Schrödinger's wave equation as more than just waves, but probability outcomes.
This exceedingly frustrated Schrödinger, and so like his ally Einstein, came up with a thought experiment to disprove the probability interpretation of his wave equation once and for all. So he came up with the though experiment of Schrödinger's cat (Einstein came up with these thought experiments all the time).
Schrödinger's cat was supposed to disprove the quantum nature of matter, because it was to be interpreted to be obsurd (how can a cat be both dead and alive?). But in the decades since, it has served to more demonstrate the strangeness of quantum systems. Quantum mechanics, much to the chagrin of Schrödinger, was and is still here to stay. Every prediction it has made has been proven so far. Einstein never liked it, he thought that someday it would be revealed as part of a bigger theory - he didn't like God throwing the dice.. D:
HTH