A concept in quantum mechanics and in quantum field theory,
which is balked at by many, when usually there's no reason for the
slightest of double takes. Truly, each of the two quantizations
hinted at by the name of this concept are familiar to any physicist,
nay to any undergrad. They are very used to working with them in
completely classical physics, except that they are used to see each
show up in a different situation, and they think of them as entirely
intuitive. But when the two show up together when one starts studying
quantum mechanics and then quantum field theory, it does feel a little
weird and counter-intuitive. I will try to present the two
quantization as each comes up in its intuitive sense, and then see how
each quantization enter into the description of the other problem.

Consider the classical problem of standing
waves on a stretched piece of string clamped at both ends (or any
other equivalent situation that satisfies the same
equations). The motions allowed can be broken up by frequency, but
not all frequencies are represented. Only a discrete set of normal
modes turn out to exist. This is first quantization -- the modes of
oscillation are discrete. If you say "Why do you call this
quantization? This isn't a quantum phenomenon, this
is classical!" you are right, but only because
that is how we've come to use the adjectives "quantum" and "classical".
If semantics were up to me, I would call this a quantum phenomenon.

Indeed, when the waves involved are not string waves but the
wavefunctions of a particle in a container ("box" is the
technical term), and when we refer to normal modes of the
wavefunction for some reason by the crazy name of "quantum states",
people come to call the exact same phenomenon as above the quantization
of states. But now suppose we want to put more than one particle in our
box (assume that the particles do not interact), then one can think of
each of the modes, or "states", as being occupied by a certain number
of particles -- zero particles, one particle, two particles, and so on.
Do you wonder why we can't have a "state" be occupied by six-and-a-half
particles? Or any continuous number, for that matter? No, I
didn't think you did, because it's intuitive. Again, ridiculous me, I'm
going to call this result ("obviously classical") a quantum phenomenon.
It is second quantization.

And indeed, if we go back to our waves on the string (or you might
like to think of electromagnetic waves now), and I tell you that each
normal mode cannot have an arbitrarily continuous amplitude, but must
take one of a discrete list of allowed amplitudes, you would surely
think of this as non-classical. But it's the same thing, really, as the
above restriction of a whole number of particles. Here, we call these
things that come in whole numbers "photons" or "phonons" or even
"magnons" (depending on context), and we call them the quanta of
our waves.

So there, if you can wrap your mind around the equivalence of the
two situations (the waves on the string and the particles in the box)
and convince yourself that they are just different ways to think about
the same thing; that particles are really just quanta of a waving
field, that waves are just the wavefunctions associated with
"particles" like photons; in fact that particles and waves are just
shadows on different walls of Plato's cave of
the same higher-dimensional concept; that this thing they are both
projections of, and that shall remain unnamed for fear of conjuring
either of the two sets of connotations, smells as sweet by any name; then you have grokked
quantum field theory. Congratulations!