Motivation for spinors 
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*After it was found that the gamma matrices couldn't be Pauli matrices and only had to be larger and even, why was their need to define a new algebraic object (i.e a Dirac spinor)? 

*Why couldn't a vector or a tensor do? 

*Plus what exactly is a spinor as compared to vectors?
 A: As I had wrote here, the Lorentz group can be represented as the direct product of two $SU(2)$ or $SO(3)$ groups. The $SU(2)$-realization of the irrep, $\left( \frac{n}{2}, \frac{m}{2}\right)$, refers to the spinor tensor $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$, where the sum $n + m$ represents the spin value of representation: $\frac{n + m}{2} = s$. If $s$ is half-integer, we must work with spinor indices. If $s$ is integer, we may convert all spinor indices to vector one and forget about spinors.
The one-particle representations of the Poincare group (which describes of particle with definite mass and spin/helicity) are also characterized by spinor tensors $\psi_{a_{1}...a_{n}\dot{b}_{1}...\dot{b}_{m}}$, which are the irrep of the Lorentz group. So we assume that at least for describing the each free spin-half particle we must work with spinor indices.
The Stern-Gerlach experiment had showed that electron has spin one half. So as the Poincare irrep with spin $\frac{1}{2}$ and mass $m$ it must be described by 2-spinor $\psi_{a}$ (or by the irrep $\left(\frac{1}{2}, 0\right)$ ) of by 2-spinor $\psi_{\dot {a}}$ (or by the irrep $\left(0, \frac{1}{2}\right)$) . But it is not hard to show, that this representation isn't invariant under $C, P, T$-transformations (while the real theory is invariant), so we must to take the direct sum of $\left(\frac{1}{2}, 0\right)$ and $\left(0, \frac{1}{2}\right)$. Corresponding object is called Dirac spinor.
Moreover, all half-spin free particles in $C, P, T$-invariant theory are described as object like Dirac spinor which satisfies the Dirac equation (look here). 
