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We're now starting to talk about quantum mechanics, the physics of the very small.
At the end of the 19th century one of the most intriguing puzzles in physics involved the spectrum of radiation emitted by a hot object. Specifically, the emitter was assumed to be a blackbody, a perfect radiator. The hotter a blackbody is, the more the peak in the spectrum of emitted radiation shifts to shorter wavelength. Nobody could explain why there was a peak in the distribution at all, however; the theory at the time predicted that for a blackbody, the intensity of radiation just kept increasing as the wavelength decreased. This was known as the ultraviolet catastrophe, because the theory predicted that an infinite amount of energy was emitted by a radiating object.
Clearly, this prediction was in conflict with the idea of conservation of energy, not to mention being in serious disagreement with experimental observation. No one could account for the discrepancy, however, until Max Planck came up with the idea that a blackbody was made up of a whole bunch of oscillating atoms, and that the energy of each oscillating atom was quantized. That last point is the key : the energy of the atoms could only take on discrete values, and these values depended on the frequency of the oscillation:
Planck's prediction of the energy of an oscillating atom : E = nhf (n = 0, 1, 2, 3 ...)
where f is the frequency, n is an integer, and h is a constant known as Planck's constant. This constant shows up in many different areas of quantum mechanics.
The spectra predicted for a radiating blackbody made up of these oscillating atoms agrees very well with experimentally-determined spectra.
Planck's idea of discrete energy levels led Einstein to the idea that electromagnetic waves have a particle nature. When Planck's oscillating atoms lose energy, they can do so only by making a jump down to a lower energy level. The energy lost by the atoms is given off as an electromagnetic wave. Because the energy levels of the oscillating atoms are separated by hf, the energy carried off by the electromagnetic wave must be hf.
Einstein won the Nobel Prize for Physics not for his work on relativity, but for explaining the photoelectric effect. He proposed that light is made up of packets of energy called photons. Photons have no mass, but they have momentum and they have an energy given by:
Energy of a photon : E = hf
The photoelectric effect works like this. If you shine light of high enough energy on to a metal, electrons will be emitted from the metal. Light below a certain threshold frequency, no matter how intense, will not cause any electrons to be emitted. Light above the threshold frequency, even if it's not very intense, will always cause electrons to be emitted.
The explanation for the photoelectric effect goes like this: it takes a certain energy to eject an electron from a metal surface. This energy is known as the work function (W), which depends on the metal. Electrons can gain energy by interacting with photons. If a photon has an energy at least as big as the work function, the photon energy can be transferred to the electron and the electron will have enough energy to escape from the metal. A photon with an energy less than the work function will never be able to eject electrons.
Before Einstein's explanation, the photoelectric effect was a real mystery. Scientists couldn't really understand why low-frequency high-intensity light would not cause electrons to be emitted, while higher-frequency low-intensity light would. Knowing that light is made up of photons, it's easy to explain now. It's not the total amount of energy (i.e., the intensity) that's important, but the energy per photon.
When light of frequency f is incident on a metal surface that has a work function W, the maximum kinetic energy of the emitted electrons is given by:
Note that this is the maximum possible kinetic energy because W is the minimum energy necessary to liberate an electron. The threshold frequency, the minimum frequency the photons can have to produce the emission of electrons, is when the photon energy is just equal to the work function:
Although photons have no mass, they do have momentum, given by:
Convincing evidence for the fact that photons have momentum can be seen when a photon collides with a stationary electron. Some of the energy and momentum is transferred to the electron (this is known as the Compton effect), but both energy and momentum are conserved in such a collision. Applying the principles of conservation of energy and momentum to this collision, one can show that the wavelength of the outgoing photon is related to the wavelength of the incident photon by the equation:
To explain some aspects of light behavior, such as interference and diffraction, you treat it as a wave, and to explain other aspects you treat light as being made up of particles. Light exhibits wave-particle duality, because it exhibits properties of both waves and particles.
Wave-particle duality is not confined to light, however. Everything exhibits wave-particle duality, everything from electrons to baseballs. The behavior of relatively large objects, like baseballs, is dominated by their particle nature; to explain the behavior of very small things like electrons, both the wave properties and particle properties have to be considered. Electrons, for example, exhibit the same kind of interference pattern as light does when they're incident on a double slit.
In 1923, Louis de Broglie predicted that since light exhibited both wave and particle behavior, particles should also. He proposed that all particles have a wavelength given by:
Note that this is the same equation that applies to photons.
de Broglie's prediction was shown to be true when beams of electrons and neutrons were directed at crystals and diffraction patterns were seen. This is evidence of the wave properties of these particles.
Everything has a wavelength, but the wave properties of matter are only observable for very small objects. If you work out the wavelength of a moving baseball, for instance, you will find that the wavelength is far too small to be observable.
All these ideas, that for very small particles both particle and wave properties are important, and that particle energies are quantized, only taking on discrete values, are the cornerstones of quantum mechanics. In quantum mechanics we often talk about the wave function 
of a particle; the wave function is the wave discussed above, with the probability of finding the particle in a particular location being proportional to the square of the amplitude of the wave function.