Why do we care about chirality? I'm trying to figure out what's the importance of chirality in QFT. To me it seems just something mathematical (the eigenvalue of the $\gamma^{5}$ operator ) without any physical insight in it. So my question is why do we care about Chirality, and why is it important?
 A: What distinguishes the chiral projections from other projections is that they are invariant under (commute with) continuous Lorentz transformations. Therefore, it's possible to have a theory in which only one projected "half" of the spinor exists, but which is still Lorentz invariant. That's interesting. And the real world turns out to be that way, so it's also important.

To motivate the projections geometrically, let me start with an easier case, which is rotation in 4+0 dimensions.
In 4 or more spatial dimensions, it's possible to have simultaneous independent (commuting) rotation in two perpendicular planes.
A general rotation in four dimensions is a rotation by different angles (which may be zero) in perpendicular planes. However, you can always write any rotation as a composition of two rotations by equal angles in perpendicular planes. A rotation by $θ$ in the $wx$ plane and $φ$ in the $yz$ plane is a rotation by $(θ+φ)/2$ in the $wx$ and $yz$ planes composed with a rotation by $(θ-φ)/2$ in the $wx$ and $zy$ planes, where I've reversed the order of $zy$ so that the rotation is in the opposite direction.
Furthermore, as in three dimensions, orthonormal bases in four dimensions can be classified as right- or left-handed, and you can classify equiangular rotations as right- or left-handed by whether the rotational planes "concatenated" ($wxyz$ or $wxzy$) form a right- or left-handed coordinate frame. Then every rotation decomposes not just into two equiangular rotations, but into one right-handed and one left-handed equiangular rotation.

In the case of massless fields in 3+1 dimensional spacetime, there is rotation in the 2D plane perpendicular to the direction of propagation, and there is also rotation in internal dimensions corresponding to the gauge forces.
In the simplest case, $U(1)\cong SO(2)$, there is one internal plane of rotation, so when you combine it with the spatial rotation you have two perpendicular planes, and you can decompose the rotation into equiangular rotations of opposite handedness as before. It's possible for only one of the two to exist. Conventionally, the internal rotation is represented by a complex phase; that's the reason there is a factor of $i$ in $γ^5$. There is no chiral decomposition without the internal gauge field, but in 5+1 dimensions there would be, since you would have four spatial dimensions perpendicular to propagation.
This simple chiral gauge theory is largely academic since the only physically relevant $U(1)$ gauge theory is QED, which isn't chiral.
The full gauge group of the Standard Model has a complicated structure, but it can be embedded in SO(10), the group of rotations of 10-dimensional space. (Actually in Spin(10), but that's out of scope for this answer.) You therefore have in total 12 dimensions perpendicular to propagation, and rotation in 6 perpendicular planes, one external and 5 internal. It is still possible to decompose the rotation into right- and left-handed parts, though the geometric interpretation of this in more than 4 dimensions is not clear to me. It's possible for only one handedness to exist without breaking continuous Lorentz invariance, and that turns out to be true in reality.
A: 
Parity involves a transformation that changes the algebraic sign of the coordinate system. Parity is an important idea in quantum mechanics because the wavefunctions which represent particles can behave in different ways upon transformation of the coordinate system which describes them. Under the parity transformation:



The parity transformation changes a right-handed coordinate system into a left-handed one or vice versa. Two applications of the parity transformation restores the coordinate system to its original state.


It is a reasonable presupposition that nature should not care whether its coordinate system is right-handed or left-handed, but surprisingly, that turns out not to be so. In a famous experiment by C. S. Wu, the non-conservation of parity in beta decay was demonstrated

.........

subsequent experiments have consistently shown that a neutrino always has its intrinsic angular momentum (spin) pointed in the direction opposite its velocity. It is called a left-handed particle as a result. Anti-neutrinos have their spins parallel to their velocity and are therefore right-handed particles. Therefore we say that the neutrino has an intrinsic chirality.

"Chiral" is an adjective coming from the ancient greek word meaning "hand" (χείρ).
When the observed handedness of specific particles and interactions came to be described with the mathematics of Quantum Field Theory,  the "chiral" adjective was chosen, instead of "handedness". The physical insight is that having it in QFT allows the correct modeling of the data by QFT.
A: I'll give a slightly more mathematical answer. First let me expand upon what chirality is all about. Quantum fields transform on specific representations of the Lorentz group. The irreducible representations are known as the $(A,B)$ representations and they are labelled by two integers or half-integers $A$ and $B$. If you have never seem this, please see Weinberg's The Quantum Theory of Fields Chapter 5.
The representations $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$ are the Weyl representations. Fields transforming on those representations have spin $\frac{1}{2}$, thereby being fermions, and are known as chiral fermions. The ones in the $(\frac{1}{2},0)$ representation are called left-handed Weyl fermions and the ones in the $(0,\frac{1}{2})$ representation are called right-handed Weyl fermions. It is possible to show that these can be taken as the building blocks for all other fields, so mathematically they are already quite relevant.
The chiral fermions have the peculiar property that they are necessarily massless particles. The reason for that is that to build a Lorentz invariant mass term in a Lagrangian density it takes both an object from the $(\frac{1}{2},0)$ representation and another from the $(0,\frac{1}{2})$ representation. A standalone chiral fermion does not admit a mass term!
On the other hand the standard Dirac fermions, like the ones we encounter in QED, have mass. The Dirac field transforms in the representation $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ and therefore it can be understood as a composite object formed out of two chiral fermions. This is one particular instance of what I have said that they are building blocks for all representations. This is where the $\gamma^5$ story comes in: the chiral projectors applied to a Dirac field project onto the irreducible left or right handed Weyl representations.
Now making more contact with what we find in the real world, consider the Standard Model. It is a gauge theory with gauge group ${\rm SU}_{\rm C}(3)\times {\rm SU}_{\rm L}(2)\times {\rm U}_{\rm Y}(1)$. This theory is built from the ground up with chiral fermions. Consider for instance the leptons in the electroweak sector ${\rm SU}_{\rm L}(2)\times {\rm U}_{\rm Y}(1)$. One such lepton would be the electron. Since we know in the real world leptons have mass, we assign to each lepton two chiral fermions $\ell_L$ and $\ell_R$. But for each lepton we have one associated neutrino and the neutrino in the Standard Model, does not have mass. Indeed the neutrino of lepton $\ell$ gets just a single chiral left-handed fermion $\nu_\ell$.
Now do you see that little ${\rm L}$ I have written down on the ${\rm SU}(2)$ group? It is to remind ourselves that the left-handed part of each lepton, $\ell_L$, and the associated neutrino $\nu_\ell$, make up one ${\rm SU}(2)$ doublet $L_\ell=\begin{pmatrix}\nu_\ell \\ \ell_L\end{pmatrix}$. These fields are charged under ${\rm SU}(2)$, while the right-handed part of the lepton, $\ell_R$, is one ${\rm SU}(2)$ singlet, and therefore is neutral under ${\rm SU}(2)$.
In that setting, an explicit mass term for the lepton would couple $\ell_L$ and $\ell_R$ and this would be incompatible with the symmetry we have. In the end of the day, the Higgs mechanism gives rise to a mass term in the phase of broken symmetry, through the Yukawa coupling of the lepton to the Higgs field. Of course the Higgs field has the right quantum numbers so that the coupling is indeed symmetric. After all, in Spontaneous Symmetry Breaking the Lagrangian is symmetric and the symmetry is broken by the vacuum, which is not invariant.
Of course, it's not possible to give a comprehensive overview of the electroweak theory in a single answer, but I hope these brief remarks make it clear that chirality has a strong presence in the Standard Model.
Still regarding the Standard Model, there is also the QCD sector ${\rm SU}_{\rm C}(3)$ of the Standard Model, in which chiral symmetry plays a big role in the discussion of mass generation, as I have mentioned in comment.
Finally I would also like to say that in supersymmetry chiral fermions are quite natural. There are two equivalent formalisms for SUSY: one using chiral fermions, which we find for example in Wess & Bagger, and another using Majorana fermions. Since they are equivalent this ends up being a matter of taste. Personally, I find the formalism with chiral fermions to be more elegant and nicer to manipulate.
