What exactly is a quantum of light?

I am currently trying to learn some basic quantum mechanics and I am a bit confused. Wikipedia defines a photon as a quantum of light, which it further explains as some kind of a wave-packet.

What exactly is a quantum of light?

More precisely, is a quantum of light meant to be just a certain number of wavelengths of light (something like "1 quantum = a single period of a sine wave" perhaps?), or is the concept completely unrelated to wavelengths? In other words, how much is a single quantum?

Answer

There are two meanings usually attached to the word "quantum" in quantum theory, one colloquial and one technical.

As you know, electromagnetic radiation behaves in ways characteristic of both waves and particles. For non-specialists, it's easy to think of a particle as being a "unit" of the wave, and since "quantum" means a unit of something, the word has gotten associated with "particle." But in reality, the idea of a particle isn't precisely defined. When people talk about a particle of light, the EM field associated with what they probably mean could be described as a wave packet, which you can think of as an electromagnetic wave that is localized to some small region in space. For example, something like this:

This is just an example, of course; wave packets can have all sorts of shapes.

The more precise, technical meaning of "quantum" has to do with Fourier decomposition. As you may know, any function can be decomposed into a sum of sine waves (or complex exponentials),

$$f(x) \propto \int e^{ikx}\tilde f(k)\mathrm{d}k$$

For any given momentum $k$, the amplitude $\tilde f(k)$ represents the contribution of the sine wave with that frequency to the overall wave. Now, classically the value of $[\tilde f(k)]^2$ at each $k$ represents a bona fide contribution to the energy of the light. But the assumption that makes quantum theory quantum is that $[\tilde f(k)]^2$ instead represents the probability that there is a contribution to the energy of the light coming from that frequency. The actual contribution that can come from any given frequency can only be one of a set of specific values, which are integer multiples of some unit $\hbar c/k$. "Quantum" is the word for that unit of energy.