First, the particle-like nature of light energy. Planck proposed the quantization of energy (E = hn) in 1900, and Einstein used it to explain the photoelectric effect in 1905. In doing so, Einstein pointed out that light must be emitted in discrete, particle-like chunks having each a particular direction rather than in wave-pulses which each spread in every direction.

Second, the nuclear model of the atom. Rutherford discovered the atomic nucleus in 1911, and this immediately created a problem. Electrons were already known; if they orbited the nucleus, why didn't they fall into it?

Bohr realized that the quantization of energy could be used to explain why the nuclear atom didn't collapse, and did so with two postulates:

1. The Existence of Stationary States. An atom exists in any one of several stationary states, each corresponding to a definite value of the energy E of the atom; and transition from one stationary state to another is accomplished by release or absorption of energy equal to the energy difference between the two stationary states.

2. The Bohr Frequency Rule. The frequency n of the radiation emitted or absorbed by a system changing between two energy states E₁ and E₂ is given by

   hn = |E₁ - E₂|

He used these postulates to explain the spectrum of the hydrogen atom, which no one had been able to explain successfully before.

My source is Pauling and Wilson, Introduction to Quantum Mechanics with applications to chemistry, now published by Dover Publications.