What were the observations and inconsistencies that lead to the development of the equation? Why was the eqution needed? What purpose did it fill?
In the orbital model of the atom, if you were to detect where electrons were in their orbits (by firing other particles at the atom I guess) you would find the electrons in a specific place. In real observations you instead find electrons all over the place, but with a higher probability of finding them in specific places, and a zero probability of finding them in other specific places.
Wave-particle duality: Firing electrons through double slits produces an interference pattern, even when there is only one electron at a time passing through the slits.
Pilot wave theory says there is a wave that the particle "rides" on that determines where the particle will go. In this way the waves produce the interferrence pattern in the double slit and the particles follow those waves to make the interference pattern.
"The de Broglie hypothesis helped explain these phenomena by noting that the only allowed states for an electron orbiting an atom are those that allow for standing waves associated with each electron."
The Bohr model came from the de Broglie hypothesis.
"The wave and particle nature of electrons was experimentally observed in 1927, two years after the discovery of the Schrödinger equation" - how?
If you go to the beach you might see a wave coming into shore. It looks like something is travelling towards you when in fact the water molecules are just moving up and down (or in a circular motion). So what is it that is moving towards you?