Undergrad Pauli spin matrices and Eigen spinors

[Rahulrj]

So I have been studying the case of spin 1/2 and I have understood how the formulations work through to find the spin matrices. However I do not get an intuitive understanding of what they mean and why they are formulated the way they are. I follow Griffith's book and in it as he begins to solve the case for spin 1/2, he writes a general state for it which he calls a spinor and writes this
$[a\,b]^T =aX_++bX_{-}$
My first doubt is what does this general state mean? Isn't there only two states of spin, spin up and spin down?
and to find the matrices of $S^2$ and $S_z$ he uses their eigenvector equation to find that but why is the same method not employed for finding the matrices for $S_x$ $S_y$ ? He gets their respective matrices using a sum of raising and lowering operators Also he ends up with an eigen spinor with a different value. So am pretty much confused with how they came to be.

 

[atyy]

There are not only two states of spin. The up state and the down state together form just one possible set of basis vectors. An arbitrary state is a linear sum or superposition of the basis vectors.

Spin is defined analogously to angular momentum http://farside.ph.utexas.edu/teaching/qmech/Quantum/node87.html