Why do we say that in non-relativistic limit we need only two component spinor? Why do we say that in non-relativistic limit we need only two component spinor?
(As in Schrödinger equation, we do not even talk of spinors,... they are one component object) I have read this statement in several books discussing non relativistic limit of Dirac equation.
 A: The term "Schrodinger equation" is ambiguous, and can sometimes refer to the abstract equation $H|\psi> = E|\psi>$ and sometimes refer to more specific things such as the spacial portion of the non-relativistic wave-function.  The non-relativistic wave-function of a particle with spin does involve spinors, so the general Schrodinger equation applied to it has both a spacial part and a spin part.
It's true that the spinors you need to describe a non-relativistic particle are only 2-component rather than 4-component, as long as the particle's kinetic energy is much less than its rest energy ($\frac{p^2}{2m}  << mc^2$)
The easiest way I can think of to explain the difference is in terms of anti-particles.  Non-relativistic quantum mechanics only describes particles, there is no such thing as anti-particles.  So it makes sense that you only need half the number of degrees of freedom.  The same 4-component Dirac spinor in quantum field theory is used to represent both the electron and the positron.  Another way to say this is that there are both positive-energy and negative-energy solutions to the Dirac equation.  But in non-relativistic quantum mechanics, there is no reason to consider antiparticles or negative-energy solutions.  We only care about describing electrons (or other similar particles).  If you take the non-relativistic limit of a 4-component Dirac spinor, you just end up with 2 of the components always being the same, and the other 2 components always being the same.  So you may as well just discard the duplicated parts and only use 2-components.
In terms of group theory, the spacetime group for relativistic quantum mechanics is the Lorentz group SO(3,1).  When you look at all representations of that, you find a trivial scalar representation, some spinor representations, some vector representations, etc.  The spinor representations are better thought of as SU(2)xSU(2), which is locally isomorphic to SO(3,1).  Since there are 2 copies of SU(2), you have two 2-component Weyl spinors (which can be stacked on top of each other to build a 4-component Dirac spinor).  But in non-relativistic quantum mechanics, time is not a dimension, it's just a separate parameter.  The group of rotations in 3D space is just SO(3).  So you only need one copy of SU(2) to find a spinor representation that's locally isomorphic to that.  Adding a second copy would just be redundant.  This means you only need a single 2-component spinor.
