Where it all started in the early part of the 20th century, with the work of Max Planck. Classical physics made the absurd prediction that a black-body would radiate an infinite amount of energy, which quite obviously did not fit with observation. In 1900, Planck proposed that the energy of electromagnetic radiation is proportional to integral multiples of the frequency of the radiation. The constant of proportionality became known as Planck's Constant. Planck suggested that energy must occur in some multiple of a very small quantity. Because this quantity is so small, the fact that energy occurs in these packets is never noticed in every day life. This is similar to how the individual pixels on a computer monitor are not normally seen unless one looks very closely. Einstein went on, in 1905, to explain the photoelectric effect by proposing that light comes in packets that he called "quanta". Later, they became known as photons.
J. J. Thomson and his team discovered the negatively charged electron in 1897. This led in 1904 to Thomson's "Plum Pudding Model" of the atom where electrons were embedded in a soup of positive charge. The atomic nucleus had yet to be discovered. Following further experiments, in 1911 Rutherford suggested that the atom was more like a "miniature solar system", with electrons orbiting the central nucleus. Unfortunately, it was shown that in this model the atom would be unstable as the orbiting electrons would emit electromagnetic radiation, lose energy and spiral into the nucleus within a fraction of a second. Niels Bohr proposed an amended model in 1913 where electrons could take only certain orbits, and to move from one to another would either absorb or emit a specific "quantum" of energy; a photon, in fact. This proposal also predicted many more spectral lines than had been seen, and once these were identified experimentally, the Bohr model for atoms was accepted.
The problem with the Bohr model was that the electron orbits were circular in a two dimensional plane. Nowadays, this model is modified so that electrons are visualized as an "orbital" which is a three dimensional shell or "cloud" representing the possible locations for the electron; essentially, a probability distribution of its' location. The orbitals are either spherical or lobular. The need to absorb or emit a photon to move from orbital to orbital remains.
In 1924, De Broglie proposed the idea of wave/particle duality to explain how an electron, in the same way as a photon, appeared to behave as a particle, sometimes, and a wave, at other times. It was not until 1927 that George Thomson, of Aberdeen University in Scotland, observed the predicted interference patterns when he passed a beam of electrons through a thin metal film. Schrödinger extended this idea in 1925, and came up with a wavefunction which is called the "Schrödinger equation", and heralded the start of the New Quantum Mechanics.
The period from Planck's work in 1900 up to 1925 is usually referred to as the "old Quantum Theory", which contained a number of inconsistencies and paradoxes.
Old Quantum Mechanics