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?
 A: Just a remark that might be helpful to understand what photon is: the "wavelengths of light" seems to be just a theoretical value calculated with the help of Planck model. What can really be measured in the experiment is the momentum/energy of photon, not the wavelength. For instance, the "colour" of the photon is fully determined by its momentum. 
A: 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.
A: A quantum of light is a particle of light which can disappear, giving its energy to an atomic or particle system, or appear, taking energy away from a particle or atomic system. A quantum of light of wavelength $\lambda$ is the minimum amount of energy which can be stored in an electromagnetic wave at that wavelength, which is Planck's constant h times the frequency. The photon is not related to the wave in any concrete way, the classical wave is a superposition of a large number of photons which are coherent.
A: Here are some things that might help:
Everything has wave-particle duality (even us). This 'effect' is not limited to the scale of single particles (microscopic/subatomic scale) like electrons. By the correspondence principle in quantum mechanics these quantum phenomena map onto the macroscopic scale (this can be loosely thought of as the scale of the world that we exist in).
Getting closer to answering your question:
Light (or in general EM radiation) propagates through space as a wave, but it interacts with matter as a particle which we call photons. The Photoelectric effect experimentally showed this (by accident in fact) and in 1905 Einstein provided the proof. Louis de Broglie in fact shown that if waves can behave as particles then particles can behave as waves.

What exactly is a quantum of light?

I won't say anything on the wavepacket explanation as this has already been explained in detail in another answer.
But a quantum of light is often thought of as a discrete amount of energy that the photon of light can have. That is to say the energy is quantized and no longer continuous. So the photons themselves have energy quanta. 
