Quantum mechanics represents such a break from classical [Newtonian] mechanics and is still -- 100 years later -- so anti-intuitive, that we pause to look at three experiments/observations that demonstrated the cracks arising back then in classical physics: blackbody radiation, the photoelectric effect and atomic spectra.
At room temperature a blackbody [that is, a body which reflects no incident light] emits infrared light but, as it is heated, this radiation moves into the visible spectrum, beginning at orange and increasing towards bluish white. Using classical theories, Rayleigh and Jeans derived an equation for the spectral density (energy density per unit frequency per volume) as a function of frequency. This model, however, was a horrible failure in that it did not correlate at all with experimental results, leading instead to a spectral density that reaches infinity in the ultraviolet regime [leading to what we now call the ultraviolet catastrophe].
Max Planck, in 1900, solved the problem (and unwittingly triggered the quantum revolution) by assuming that the blackbody was made of oscillators that could emit energy only in packets of nhν, where n is a nonnegative integer and h is a constant (now called Planck's constant). This assumption leads directly to the Planck distribution, which fit the experimental data perfectly when h=6.626E-34 Js. He spent the next few years unconvinced that h had much physical meaning and was not a strong supporter of the quantum theory, but by the time Einstein and Bohr hopped aboard, the train had left the station.
The photoelectric effect, in which light shined onto a metal induces a current [but only if its frequency is above some threshold], was also a mystery awaiting an explanation. Einstein's radical 1905 postulate, for which he won the Nobel Prize in Physics, was that light was comprised of corpuscular [particle-like] packets of energy hν called photons. The photoelectric effect could then be understood as a simple collision between two particles -- the photon and the electron in the metal. If the frequency of light were high enough, it would have an energy sufficient to overcome the binding energy [called the work function] of the electron to the metal. Einstein also, to his chagrin years later, the idea of the wave-particle duality, which is one of the more fundamental and vexing aspects of quantum mechanics.
When the emitted light of a heated gas is separated through a prism, a line spectrum occurs [as opposed to the continuous rainbow-type spectrum seen from sunlight]. The wavelengths of these lines could be fit to the Rydberg formula but, since it is simply an empirical formula and not based on any underlying theory, it further demonstrated the problems of classical physics. The Rydberg formula was eventually theoretically derived in the work of Bohr, whose model of the hydrogen atom we will visit a little later.
Much of what we have described thus far, including the early work of Planck, Einstein and Bohr, is generally referred to as the old quantum theory. One giant step further, spurred on by the work of deBroglie, is quantum mechanics, which we will visit next.