What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric? What are $\mu$ and $\nu$ in $g_{\mu\nu}$ metric?
Consider the metric $g_{\mu\nu} = \begin{pmatrix}
1 & 0 &0 \\ 
0 & r^2 & 0\\ 
0 & 0 & r^2\sin^2\theta
\end{pmatrix}$
 A: Those Greek letters are indices indexing the components of $g$. Generally if one expresses a rank-2 tensor like $g$ as a matrix, the first index indexes the rows, the second the columns. In your example, we have $g_{rr} \equiv g_{11} = 1$, $g_{\theta\theta} \equiv g_{22} = r^2$, $g_{r\theta} \equiv g_{12} = 0$, etc.
As you can see, we sometimes use numbers to refer to specific indices, and sometimes letters related to the coordinate system in play. When you have not just a metric for space but one for spacetime, $0$ is frequently used to index the time component.
Often if we use Latin letters $i,j,\ldots$ as free indices we mean "this index is spatial, not temporal." Of course some authors reverse the roles of Greek and Latin here, just to keep you on your toes.

More pedantically, "$g_{\mu\nu}$" is an indeterminate component of $g$, whereas the matrix you wrote is the collection of all components in some understood basis. It would probably be more proper to either write
$$ g \stackrel{\mathcal{B}}{\longrightarrow}
\begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & r^2 \sin^2\!\theta
\end{pmatrix}
$$
(the tensor as expressed in some basis $\mathcal{B}$, which happens to be the standard $(r,\theta,\phi)$ of spherical coordinates in our case, is represented by the matrix), or
$$ (g_{\mu\nu})_{\mu,\nu} = \begin{pmatrix}
1 & 0 & 0 \\
0 & r^2 & 0 \\
0 & 0 & r^2 \sin^2\!\theta
\end{pmatrix} $$
(the matrix of values $g_{\mu\nu}$, ranging over indices $\mu$ and $\nu$, is equal to the matrix, where $\mathcal{B}$ is understood). But these are both rather overbearing notation, and so are rarely used.
A: $$g_{\mu\nu}=\begin{pmatrix}g_{00}&g_{01}&g_{02}\\g_{10}&g_{11}&g_{12}\\g_{20}&g_{21}&g_{22}\end{pmatrix}$$
$\mu,\nu=0,\ldots,N$ are the matrix indices of the metric (and of tensors in general) in $N+1$ dimensions.
