To quote Halmos:
If $R$ is an equivalence relation in $X$, and if $x$ is in $X$, the equivalence class of $x$ with respect to $R$ is the set of all elements $y$ in $X$ for which $x R y$. Examples: if $R$ is equality in $X$, then each equivalence class is a singleton; if $R = X \times X$, then the set $X$ itself is the only equivalence class.
~P. R. Halmos, Naive Set Theory (p. 28)
The first one, I think I understand. Each equivalence class is a singleton because each element $x$ in $X$ is only equal to itself.
The second is confusing me further the more I think about it, perhaps because of the wording. If $R = X \times X$, do I still consider it to be 'in' $X$ or is it 'in' the result of $X \times X$? If it's the former, how is that any different than equality in $X$? When comparing across sets rather than within one set, the equality results should still be the same, yielding a number of singletons. If it's the latter, then surely we're now dealing with a series of ordered pairs that did not exist in the set $X$ beforehand, precluding it from being the equivalence class.
Or, is it that it's neither of these, and the set $X$ used here is being treated like the $x$ we are seeking equivalence classes for in his initial definition? This latter definition seems to be the only way I can get my head around how $X$ itself ends up being the equivalence class, but also seems like I'm missing something vital in making that assumption.