Consider the set $A = \\{a,b,c,d,e,f,g,h,i \\}$, and look at the array: $$\begin{pmatrix} a & b & c \\\ d & e & f \\\ g & h & i\end{pmatrix}$$Define $\sim$ on $A$ by saying that $x \sim y \iff x$ and $y$ are in the same column.
This way, we have $a \sim d$, $b \sim h$, $f \sim i$, $a \
ot\sim e$, $h \
ot\sim c$, for example. I'll leave you to check that $\sim$ is an equivalence relation as an exercise. Denote by $[x]$ the set $\\{y \in A \mid x \sim y \\}$. This way we have: $$[a] = [d] = [g] = \\{a,d,g\\},$$and similarly for the other two ~~columns~~ equivalence classes (can you guess what they are?). We can partition $A$ as $[a]\cup [e]\cup [f]$, or $[d]\cup [b] \cup [i]$, etc.
In words: two elements are equivalent if they are in the same column - the class of an element is the column which contains it. Every element in $A$ is in only one column.