Every measurable quantity A is described by a Hermitian operator $\hat{A}$.
Do you want to know what a system's energy is? Apply the energy operator, this is referred to as the Hamiltonian operator, $\hat{H}$. What's the magnetization? Apply the magnetization operator, etc.
Remember that we defined our energy with the following energy function:
$$E(\textbf{s}) = -\frac{J}{k_B}\sum_{\langle ij\rangle}s_is_j + \frac{\mu}{k_B}\sum_is_i$$
We can define the associated operator to be:
$$\hat{H} = -\frac{J}{k_B}\sum_{\langle ij\rangle}\sigma^Z_i \sigma^Z_j + \frac{\mu}{k_B}\sum_i \sigma^Z_i$$
where $\sigma^Z_i$ is the
$$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$
matrix applied to the $i^\text{th}$ qubit.
For example:
$$\begin{align}
\sigma^Z_1 \ket{0} &= \ket{0} \\
\sigma^Z_1 \ket{1} &= -\ket{1}
\end{align}$$
This is seen by simple matrix-vector multiplication. Recall that we defined:
$$\begin{align}
\ket{0} = \begin{pmatrix}1 \\ 0\end{pmatrix} \\
\ket{1} = \begin{pmatrix}0 \\ 1\end{pmatrix}
\end{align}$$
If we have multiple qubits, then the subscript just tells you which qubit to apply the operator to.
$$\begin{align}
\sigma^Z_1 \ket{01} &= \ket{01} \\
\sigma^Z_1 \ket{11} &= -\ket{11} \\
\sigma^Z_2 \ket{01} &= -\ket{01} \\
\sigma^Z_2 \ket{10} &= \ket{10}
\end{align}$$