Q:

How to understand spin component operators

I am using Quantum Mechanics by McIntyre and I am trying to understand the motivation behind spin operators. 
From the text, it says that an operator corresponds to an physical observable and that the only possible results of a measurement are the eigenvalues of an operator.
To try to understand how operators are used, I skimmed a couple sections ahead, but couldn't find examples spin operators acting on kets. For example, what is the meaning of a ket mapped by a spin operator?
What is the purpose of spin operators? Can you point me towards some demonstrative examples?
Thanks in advance.

A:

The point of observable operators is that they give a prescription for calculating probability distributions of that variable. 
Look at the spin states, $\left.|\!\!\uparrow\right>$ and $\left.|\!\!\downarrow\right>$. If we consider arbitrary superpositions, $\left.|\psi\right> = \cos \theta \left.|\!\!\uparrow\right>+\sin \theta\left.|\!\!\downarrow\right>$, with $\hat O$ an operator, that has eigenvalues for the spin states $o_1$ and $o_2$ respectively,
\begin{align}
\left<\psi\left|\hat{O}\right|\psi\right> = p_1 o_1 + p_2 o_2
\end{align}
where $p_1,$ and $p_2$ are probabilities.
That is expectation values of these operators pick out the average value of these operators. This is useful because it let's one figure out all moments of an observable by just matrix algebra, and furthermore allows calculation of these same expectation values in different basis.
Note that in this picture, what $\hat{O}\left|\psi\right>$ is is irrelevant. $\hat O$ was not defined to make it represent an experiment, it was defined to calculate average values. 

