\[ \hat{S}^2 |s,m_s\rangle = \hbar^2 s(s+1)|s,m_s\rangle \]
\[ \hat{S}_z |s,m_s\rangle = \hbar m_s |s,m_s\rangle \]
For an electron:
\[ \boxed{s=\frac{1}{2},\qquad m_s=\pm\frac{1}{2}} \]
. . .
\[ |\frac{1}{2}, \frac{1}{2}\rangle =|\uparrow\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \qquad |\frac{1}{2},-\frac{1}{2}\rangle =|\downarrow\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix} \]
\[ \hat{S}_x = \frac{\hbar}{2} \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \qquad \hat{S}_y = \frac{\hbar}{2} \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \]
\[ \hat{S}_z = \frac{\hbar}{2} \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \]
. . .
Just like orbital angular momentum,
\[ \boxed{ [\hat{S}_i,\hat{S}_j] = i\hbar\,\epsilon_{ijk}\hat{S}_k } \]