Why do we need $2^\text{nd}$ quantization of the Dirac equation As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion about it. So let me start with its definition:
The Dirac equation is given by,
$$
i \hbar \gamma^\mu \partial_\mu \psi = m c\cdot \psi
$$
where the Dirac matrices $\gamma^\mu$ are defined by 
$\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu  = \eta^{\mu\nu}$ and where
$\psi$ is a "solution". 
The first deal of confusion already starts with the $\psi$'s. It seems that people freely see them as spinor valued functions or as "operator fields". 
But if I understand this correctly, seeing them as operators, is not part of the
original picture, but was later added as the so called second quantization. Right?
Now my question is the following: Why do we need this second quantization of the
Dirac equation? What experiments can not be described by the original Dirac equation? Maybe there is a list somewhere or such?
 A: Solutions of the Dirac equation were originally interpreted as multi-dimensional wave functions or states. Each component is similar to good ol' non-relativistic quantum mechanics. This non-operator theory is sometimes called relativistic quantum mechanical spinor theory.
Yes, second quantization is a method that, after all is said and done, requires the solutions to be interpreted as operators rather than states. This is because we impose particular commutation relations among the players of the solutions, which would simply commute if they were states/wave functions. This new theory is called QFT (for spin-half particles).
One disadvantage of the non-operator theory is that some states have negative energy. Apparently that's bad. The operator-valued QFT theory, on the other hand, has all positive energies.

Source: If you don't mind paying for textbooks, this is a particularly self-study friendly one.
A: My understanding is that the original Dirac equation can only describe the state of a single relativistic fermion, while the second-quantized version can be used to define multi-fermion states. See for example section 4 of the paper at the following link: 
http://www.cond-mat.de/events/correl13/manuscripts/koch.pdf
That said, when particle physicists speak of second quantization, they usually mean quantization of a classical field. In classical electrodynamics, the electromagnetic field determines the forces exerted on a charged particle, such as an electron. In ordinary quantum mechanics, the electron is quantized-- it is described by a state or wavefunction-- while the electromagnetic field is not-- it is still just an ordinary function of space and time. In this context, second quantization involves promoting the electromagnetic field to an operator. In order to do this, one must build special relativity into the theory as well. For more details, see Peskin and Schroeder's book on quantum field theory.
A: The Dirac Equation is a relativistic wave equation of a $1/2$ spin. Surprisingly, this equation give to us a positive definite norm $\psi^\dagger\psi$, when $\psi$ is a bi-spinor:
$$
\psi = (\psi_+,\, \psi_-)^T
$$
The $\psi_\pm$ is a $\pm$ spinor.
We know that (Special Relativity)+(Quantum Mechanics), makes the density of particles and total number of particles incompatible. If we are probing length close to the Compton length of the electron, we are sensible to this effect.$$[N,\,\int_{L^3\sim\lambda_c^3}\rho(x)dx]\neq0$$$$\lambda_c=\left(\frac{\hbar}{mc}\right)$$ In the case of electrons, the first thing that we feel is the creation and annihilation of positrons-electrons pairs that give us corrections of $\Delta E\sim \alpha^4mc^2$. After this, more deep in compton length, precisely $L\sim\lambda_c^3$, the electromagnetic field makes your quantum manifestation as well as the interactions of the positrons-electrons pairs via electromagnetic field. In hydrogen atom, this is the Lamb Shift and the correction of the gyromagnetic ratio, respectively. Corrections in energy $\Delta E\sim \alpha^5 mc^2$.
$$
\alpha =\frac{1}{mc^2}\frac{1}{4\pi\epsilon_{0}}\frac{e^{2}}{\left(\frac{\hbar}{mc}\right)}=\left(\frac{1}{137}\right)
$$
The Dirac Equation can probe only $L\sim\alpha^2\lambda_c$, giving corrections $\Delta E\sim\alpha^4mc^2$. After that, the equation fail. Actually, the Dirac Equation work in terms of two spinors that can't be divided in the presence of an four vector potential $A_\mu$. In free case, we can divided by an exact Foldy Wouthuysen Transformation. In the presence of the potential, this transformation can only be done approximately, but is only interesting until $L\sim\alpha^3\lambda_c$, when the Dirac equation starts to be wrong. This transformation help us to find two equations, each for each spinor, taking average over the pairs production (average in $L\sim\alpha^3\lambda_c$). In the case of the hydrogen atom, only one equation has bound states. This equation describes the physical electron (two positive charge, proton and positron don't form bound state).
The Dirac equation describes the incompatibility of number and density of particles, but takes the EM field as classical, and neglect the interactions of electrons-positrons pairs. Only the interference of positron/electron are accounted as quantum relativistc effect. Note that what is actually positive definite is $\psi^\dagger\psi=\psi_+^\dagger\psi_++\psi_-^\dagger\psi_-=\psi_{electron}^\dagger\psi_{electron}-\psi_{positron}^\dagger\psi_{positron}$
Solving the Dirac Equation exactly is simple for a central potential. In terms of QFT - the correct way of doing relativistic quantum mechanics - the solution of Dirac equation give to us good basis of creation and annihilation operators: creation and annihilation of eigenstates of Dirac Equation. The QED calculations can be taken in terms of perturbations on number of loops in this basis.
A: When I was in the last year of graduate school back in 1963 I took a semester course on quantum field theory, creation and annihilation operators galore, and lots of theorems. I was as bemused as you, the book was Bogoliubov and we had become experts in manipulating creation and annihilation operators. Then I attended a CERN summer school and the lecture by M.Veltman brought everything to focus, why we were running around with creation and annihilation operators. 
"Weak Interactions of non strange particles " solved the riddle. It was all about calculating cross sections. Hurray, there was physics in the madness :).
So this small story of mine is to illustrate that in order to calculate cross sections before the advent of Feynman diagrams and the corresponding second quantization, setting up the integrals to calculate cross sections and compare them with experiment was a long drawn out process. (At a workshop, much later in 1980, I heard from Feynman himself how his use of Feynman diagrams allowed him to cut down the time in calculations that colleagues were amazed, during the manhattan project).
Particle physics is about cross sections, all successful theoretical models must end up in numbers giving cross sections and lifetimes, that is what particle physics is about. 

Why do we need this second quantization of the Dirac equation? 

Second quantisation + Feynman diagrams simplified life.

What experiments can not be described by the original Dirac equation?

The solutions of the Dirac equation, the wavefunctions,  are used as the basis on which the integrals prescribed by the Feynman diagrams are written out and calculated. Second quantization is a meta level which simplifies calculations, in my experimentalist's opinion.
A: Here's my own little interpretation about why the Dirac field should be "quantified" (i.e be an operator instead of an ordinary wave function), without all the talk about Pauli principle, anihilation/creation operators and all other weird QFT stuff.  For me, it's enough to understand why $\Psi$ below (not $\psi$) isn't a probability amplitude.
In a relativistic setup, all fundamental particles and fields in nature should be irreducible representations of the Lorentz group.  So we could have scalar, vector, tensor and also spinor fields.
Since any field should propagate in empty space without violating causality, the field should obey a wavelike differential equation (Klein-Gordon equation if the field has inertia ; $m \ne 0$).  In the case of the electromagnetic field $F^{ab}$ (or the vector field $A^a$) propagating in empty space, it obeys the Maxwell equations, which includes the wave equation (with $m = 0$) :
$$\tag{1}
\partial_a \, F^{ab} = 0, \qquad \Rightarrow \qquad \square \, A^a = 0.
$$
In the case of a spinor field $\Psi$, it should be the Dirac first order equation, which is also including the wave equation (with a mass term) :
$$\tag{2}
\gamma^a \, \partial_a \, \Psi + i \, m \, \Psi = 0, \qquad \Rightarrow \qquad \square \, \Psi + m^2 \, \Psi = 0.$$
Now, all the fields are in principle obervables quantities, or could have observable effects (else, it's not physics !).  The electromagnetic field is not directly obervable as such, but could have effects on electrical charges (which could reveal the presence of the electromagnetic field).  In principle, the energy-momentum tensor $T^{ab}$ of the EM field is also obervable/measurable (since it's about energy, momentum, angular momentum, etc).  So the $A^a$ field should be treated as an observable in quantum mechanics, which imposes all observables to be representend by hermitian operators.
It's the same for the spinor field $\Psi$.  It is not directly obervable as such, but it could react to an electromagnetic field, and could also generate some EM field (if $\Psi$ has a charge).  Its energy momentum $T^{ab}$ is also observable, in principle (energy, momentum, angular momentum and so on).  So in Quantum Mechanics, it should be represented by an operator.
In a general case, you have a physical field $\Phi$ (indices suppressed) propagating in spacetime as an irreducible representation of the Lorentz group, thus obeying some partial differential equation of the general shape
$$\tag{3}
\mathcal{E}(\Phi, \; \partial_a \, \Phi, \; \partial_a \, \partial_b \, \Phi) = 0.
$$
If it has an energy momentum tensor $T^{ab}$ (typically depending on the squares $\Phi^2$ and $(\partial_a \Phi)(\partial_b \, \Phi)$), then it should be regarded as an observable in QM.  This implies that it should be defined as an operator object.  Not a probability amplitude.  This is a very general consequence of standard QM, and has nothing to do with "second quantification".
"Second quantification" is really an historical mistake, done at an epoch where there was a lot of confusion about the fields and particles of Nature.  It just happened that we discovered the electron firstly as a particle (i.e. interacting with measuring devices in the lab as "particles"), and "rediscovered" a bit later that it was really just another field propagating in spacetime.  Electron isn't a pointlike particle!  If you think a bit about it, fundamental pointlike particles simply doesn't make any physical sense at all.
There is no real particles out there.  Just mathematical fields (i.e. representations of the Lorentz Group constrained by the causality principle) propagating like waves and interacting with other fields like particles.
A: The original Dirac equation does not take into account the Pauli exclusion principle (https://en.wikipedia.org/wiki/Pauli_exclusion_principle), second quantization does.
A: When we quantize the electromagnetic field, the canonical approach involves expressing the field as a set of quantum oscillators in terms of the classical fields E and B. This gives us a picture of the electromagnetic field as a wavefunction in the basis of Fock states that describes photon occupation numbers in different available modes on the field.
Second quantization applied to other equations like Klein-Gordon or Dirac field is basically treating these fields in the same footing as the electromagnetic field. This means that we treat the pure Dirac equation technically as a classical electron field, where we write down the field as a set of quantum oscillators on each field mode.
Unlike the case of the electromagnetic field or the K-G fields, 2nd quantization of the Dirac field must be done with anticommutator algebra, in order to preserve the antisymmetry properties of Pauli exclusion principle.
