A set is transitive if and only if all of its elements are subsets.
Equivalently, a set $A$ is transitive if and only if:
If $A$ is transitive, then if $x$ and $A$ are connected somehow by membership (that is, $x \in y \in z \ldots \in A$), then $x \in A$.
The intersection of two transitive sets is transitive.
In set theory, transitive sets play an important role in models of ZFC. See transitive ZFC model.