Terminology: Meaning of a Quantum When people say that a photon is a quantum of light, are they talking about 2 different types of quantizations in one phrase? To quantize something, you take something continuous and make it discrete (more or less). Energy in a light wave of frequency $f$ is continuous in (1) value and (2) location. The energy is distributed over space and if you fiddle with the amplitude knob, you can make its energy vary continuously to any value that you want. To say that its energy is quantized into steps of $hf$ means that the energy of the wave cannot vary continuously as you fiddle with the amplitude knob. The energy can only bounce between different $nhf$ levels. However, this doesn't mean that the energy has to come in packets? I can imagine a wave with quantized energy $nhf$ distributed continuously over space (therefore, its energy is quantized but the energy is not spatially quantized into packets. Physically, from a classical perspective, quantizing energy doesn't make sense. However mathematically/logically, quantizing the energy of the wave but not quantizing the location of the energy is completely fine. See signal processing: discrete amplitude signal + continuous time domain, or discrete domain + continuous range, or both discrete amplitude and discrete domain in which case you have a completely digital signal). On the other hand, to say that energy hits you in packets (instead of hitting you continuously as in classical physics) doesn't imply that the energy of those packets has to be quantized.
So a photon is light energy quantized in value and energy quantized spatially? 
I ask because I think the terminology gets confusing across different sources and people. From what I gather, when people say a wave is quantized (or something along those lines), they already have wave-particle duality in mind. So in the back of their mind, they already quantized the spatial location of the energy (into particles). Therefore when they say 'quantize the wave' with the previous sentence already in mind, they specifically mean the value of the energy.  
 A: The photoelectric effect , black body radiation, and double slit single photon at a time experiments show that classical electromagnetic wave, called light, is made up of entities that can act as particles of energy h*nu.
This is the single photon at a time double slit experiment that shows that the photons  with energy=h*nu leave  point footprint on the "screen", and their probability density builds up the classical electromagnetic interference pattern expected of classical light of frequency nu.

The quantization part means that the energies are discrete, not a continuum for the specific frequency of classical light. An energy packet impinges on the screen.
This has led for the photon to be an elementary particle in the standard model of elementary particle physics.
Light is not "quantized" by being cut into pieces of energy hnu. It is more complicated than that and needs the tools of field theory to understand the model.As an elementary quantum mechanical particle the photon has a wavefunction which obeys a quantized Maxwell's equation. Classical electromagnetic waves emerge from a large ensemble of photons according the quantum field theory, and the whole system, classical and quantum mechanical , is consistent.
A: Yes.


*

*The energy is quantized when you have, e.g., confinement as in an atom, or a particle in a box (think of the modes of stationary waves in a rope, since now the particle is described by a wave function). So when such a system emits or absorbs energy, it does so in quanta of energy.

*And when you model light as photons, then, given they're particles they are, within the limitations of the uncertainty principle, spatially localized ("quantized spatial location" in OP's words).
Of course, that's a naive, non-QFT answer that's perhaps already clarifying, but leaves room for deeper explanations.
A: Yes and no. 
First the Yes: There is not a single quantization, but many quantizations. In order not to become philosophical, let's look at history.


*

*1900, Planck's law of the black body radiation describes the energy of radiation as quantized into humps of $h$. 

*1905, Einstein's photoelectric effect describes light to be spatially quantizes into photons. Each photon carries the quantized energy $h \nu$. As a side mark, this is actually not the "proof" that light is quantized, because we get the same result, if we treat light to be continuous and the solid to be quantized. The "proof" of the quantization of light is actually the Compton effect. 

*1913, Bohr's quantizes the angular momentum of atoms thereby explaining why bounded electrons do not radiate. 

*... 

*Many years later the QED introduces the so called second quantization.


Now the No: 

I can imagine a wave with energy $nhf$ distributed continuously
  over space.

Really? How does such a wave look like? Let's take the energy of a single green photon and assume, that we spread it's energy over the length of the solar system. The amplitude of this quasi-continuous wave will be too small for any detector to be measurable. 
