What is Matrix representation, character?

Example 4.21 (The coordinates of $C_{3v}$)

For $C_3$ about $z$, $D = \left(\begin{smallmatrix}\cos120^\circ & -\sin120^\circ & 0\\ \sin120^\circ & \cos120^\circ & 0\\ 0 & 0 & 1\end{smallmatrix}\right)$, of trace $2\cos120^\circ + 1 = 0$; for $\sigma_v(xz)$, $D = \mathrm{diag}(1, -1, 1)$, trace 1; for $E$, trace 3. The characters of this representation are $(3, 0, 1)$ on the classes $(E, 2C_3, 3\sigma_v)$. Every matrix is block-diagonal, a $2\times2$ block for $(x, y)$ and a $1\times1$ block for $z$: the representation splits into two smaller ones.