1. Black Body Radiation: Every physical body absorbs and emits radiation -- for example when you heat an iron rod, it emits light whose color depends upon the temperature (from red hot to white hot). However, classical Physics was unable to model the relationship between temperature and the peak wavelength of the radiation being emitted.
2. Atomic Model: According to classical electromagnetism equations, if we assume that an electron revolves around an atomic nucleus then the electron must emit electromagnetic energy which will cause it to lose energy and fall into the nucleus. But that would be the end of the universe then!
3. When light is shown on a metallic plate, it kicks out electrons in an effect called the photoelectric effect. In this effect the electrons are emitted only if the incident light has a specific frequency or higher and the average energy of electrons emitted is dependent on the frequency of the incident light and not on the intensity of the light. Furthermore, the electrons were emitted instantaneously which is also in contrast to classical theory which predicted that the emission caused by sufficient absorption of energy would require some finite amount of time.
There were other "holes" in classical theory. For example, if the Double Slit experiment was repeated for electrons, the electrons behaved like waves in that they produced an interference pattern just like waves. However, if we set up an apparatus that monitors through which slit the electron went through, the interference pattern disappeared. Thus, electrons behaved liked waves when no one was looking at them and as particles otherwise. This was very puzzling.
Planck, who was working on Black Body Radiation modeling, hypothesized that any physical body can emit radiation only in discrete packets of energy he called quanta and not continusously. Furthermore, the energy of a quanta is dependent upon its frequency by the relation E=hf. This implied that the minimum amount of energy emitted at a certain frequency will be one quanta (at a certain frequency). This allowed Planck to model the change in peak emission wavelength with respect to temperature and verify the observations in the lab.
Einstein took Planck's idea and reversed it -- Einstein hypothesized that energy is not only emitted in quanta but it is also absorbed in discrete quanta and this can explain why we need specific frequency of incident light to emit electrons in the Photoelectric effect and why the energy of the emitted electrons depends on the frequency (or in other words) energy of incident light. Einstein modeled the incident light in the form of particles he called Photons such that the energy of the photon is given by E=hf. The physical existence of photons was proven by Compton in the Compton effect which occurs when high energy X-rays are shone on metal and the emitted X-rays have lower energies in comparison.
Bohr also used the concept of "quantized or discretized or packetized energy" to build a model of the atom in which he hypothesized the nucleus at the center and electrons rotate around it in fixed, discrete orbits. When an electron absorbs an energy packet it can move to a higher orbit and when it emits the extra energy as a packet, it falls down to a lower energy orbit. However, the electron orbit change occurs in zero time as its "movement" from one orbit to another would imply continuous energy dissipation. Continuous energy levels imply that the electron can fall into the nucleus but the discretization of energy explains why this is not the case (because an electron can only have fixed energy orbits).
This gave significant support to the idea of "discretized energy" as it explained the behavior of atoms and the photoelectric effect.
However, this got Einstein thinking -- if light can be viewed as photons and have a wavelength, which is an inherently wave property, too, then what does it mean for a particle to have a wavelength?
While Einstein was busy thinking about this. De-Broglie got another idea -- if light (which we can easily think of as a wave) can behave as a particle, then, can particles also behave as waves? He based this on his observations about the electron double slit experiment. It was known that the distance between two highs (peaks) of the interference pattern produced as a result of light double slit experiment changes with respect to the wavelength of the incident light. So if light's wavelength change can cause this distance to change, then, can the distance between peaks of electron density regions in the electron double slit experiment, be used to infer that electrons (and other particles) have an associated wavelength? He used this idea to find a relation that essentially said that every particle has an "associated" wavelength inversely proportional to its momentum. He used this equation to explain the distances between the fringes of the electron double slit experiment. He reasoned that we cannot observe such waves for large particles because their associated wavelength is too small. However, this effect is significant for small-sized objects.
Although de Broglie inferred that particles have wave-like behavior due to their associated wavelength, but what exactly is "waving" in the wave associated with the particle. This was taken up by Shrodinger: He developed an equation that models the wave associated with a particle as a "wavefunction" which is essentially a function that gives the behavior of the particle in terms of space and time. His 2nd order differential equation models how the wave function of a particle would change through time and space when it is acted upon by a force similar to how Netwon's famous equation $$F=m\frac{\partial^2 x}{\partial t^2}$$ that also relates space, time, force and matter. The significance of his wavefunction was described physically by Max Born who said that the magnitude (squared) of the wavefunction of a particle is proportional to the probability of its occurrence at a certain time at a certain position in space. His wavefunction could explain why we get interference patterns for the electron double slit experiment: when an electron's wave function encounters the slits, its behavior can be described by two separate wavefunctions which can interfere each other and render alternating bands of high and low probabilities on the target screen. The calculations by the wavefunction equation match the pattern produced exactly, i.e., the number of electrons in a certain region on the target screen observed experimentally are in agreement with the probabilities predicted by the wavefunction solution of the equation.
One of the most interesting implications of the wavefunction concept is that, due to the probabilisitc nature of the wavefunction, we can only infer the probabilities of occurrence of an object at a certain time and position in space. This is in contrast to classical theories which had no such "undeterministic" implications. However, the equation could explain anything that the classical theory could and everything beyond that as well.
This equation also explained radioactivity, or the ejection of alpha particles from the nucleus of radioactive nuclei: based on the wavefunction concept an alpha particle which is a part of a heavy or unstable nucleus, has a probability of occurrence outside the nucleus. This is why an unstable nucleus will eventually decay spontaneously if we wait long enough but it is impossible to predict when a particular one would decay. This is analogous to saying that the probability of getting heads for a particular coin is 0.1 but predicting that the next coin toss will result in heads is not possible. Similarly, the concept of half-life (the time it takes half of radioactive atoms to decay) is only meaningful when dealing with a large (statistically significant) number of atoms.
To summarise, we have seen that the consequence of discretization of energy into quanta or packets is that both particles and waves can be described in terms of wavefunctions which model their probability of having certain properties (e.g., occurrence at a certain position at a certain time) and their behavior (e.g., how would the probability of occurrence at a certain position and time would change over time when acted upon by certain forces). Thus, we have agreed that apparently both waves (light) and quantum sized objects (electrons, photons) can behave as both waves and particles. This also tells us that we can only measure probabilities and not exact positions or other properties and this has the implication that it is impossible to predict the future even when all variables are accounted for. However, we are yet to understand the following:
1. Why does the interference pattern disappear when we observe which slit a particular electron went through in the double slit experiment?
2. How can these concepts be used for computing?
3. Why does the act of observation change the behavior?
4. How can we explain the three-polarizer paradox?
5. We do not see quantum effects in macroscopic objects. Why?
6. Are there any examples of macroscopic quantum effects?
More on these later. Inspired from:Khalili, Quantum: A guide for the perplexed. As part of my course on "Quantum Programming".