There are certainly examples of such non-trivial equivalence relations. For example, in graph theory, let $G$ be an (undirected) graph and define the relation $\sim$ on its set of vertices as follows:
$a \sim b$ if and only if $a$ can be reached from $b$ by traversing a finite chain of edges in $G$.
This is an equivalence relation, as can be easily shown by proving that it is reflexive, symmetric and transitive, but its definition makes no reference to any common property shared by all equivalent vertices.
Of course, as the other answers have noted, any equivalence relation $\sim$ divides its domain into equivalence classes, and it's always possible to recharacterize the relation as "$a \sim b$ if and only $a$ and $b$ belong to the same equivalence class." In the particular case above, the equivalence classes even have an established name: they're called the connected components of $G$.
But taking that characterization as the definition of $\sim$ would make no sense, since the equivalence classes are themselves defined by the relation, and so defining the relation by the equivalence classes would be circular!
As a further demonstration of its non-triviality, it may be worth noting that the relation $\sim$ defined above would not necessarily be an equivalence relation if $G$ was a directed graph: in that case, while $\sim$ is still clearly reflexive and transitive, it may or may not be symmetric. To actually obtain an equivalence relation in that case, one needs to somehow adjust the definition to force it to be symmetric, e.g. by requiring the existence of a chain of edges in both directions (in which case the equivalence classes thus obtained are the strongly connected components of the graph).