Let
$$R = \{ \left\langle {x,y} \right\rangle \in \wp (\mathbb{Z}) \times \wp (\mathbb{Z})|\exists t \in \mathbb{Z}.y = x + t\} $$
This is the equivalence class for $\{0\}$ $$\begin{array}{l} {[\{ 0\} ]_R} = \{ A \in \wp (\mathbb{Z})|\exists t \in \mathbb{Z}.A = \{ 0\} + t\} \\ = \{ A \in \wp (Z)|\exists t \in \mathbb{Z}.A = \{ t\} \} \\ = \{ \{ t\} |t \in \mathbb{Z}\} \\ \end{array}$$
I'm not sure I understand the equivalence class here. Why is it a singelton, $\{0\}$ and not $0$? Also, I don't think I understand the role of $A$ correctly in this definition.
Well, $\wp(\Bbb Z)$ means the powerset of $\Bbb Z$, i.e. the set of all subsets of $\Bbb Z$.
In particular, $A\in\wp(\Bbb Z)$ means $A\subseteq\Bbb Z$, and that's how $\{0\}$ fits in the picture.
For a subset $x\in\wp(\Bbb Z)$, we define $$ x+t :=\{a+t\in\Bbb Z\,\mid\,a\in x\}\,,$$ i.e. the relation $R$ in words would say that $xRy$ means that subset $x$ can be shifted to subset $y$, e.g. we have $\{0,3,9\}\,R\,\{-5,-2,4\}$, and, of course, $\{0\}\,R\,\{t\}$ for any $t\in\Bbb Z$...