Does the two-element set have a categorical description in the category of (finite) sets? So more or less what I ask in the title: is it possible to identify (uniquely up to bijection) the two-element set in the category of sets as an object that has a particular (categorical) property?
EDIT: Following Hanno's answer, I would like to mention that I know $2$ is a subobject classifier. What I am instead searching for here is some other property that $2$ has (if there is one) and which is not intrinsic to toposes. 

The two-element set, call it $\Omega$, represents the subset-functor: For any set $X$, there is a natural bijection $$(\ddagger):\qquad\text{Hom}_{\textsf{Set}}(X,\Omega)\ \ \cong\ \ \text{Subset}(X).$$
Because of this, it is called a Subobject Classifier. By the Yoneda-Lemma, an object is determined up to unique isomorphism by the datum of a natural isomorphism $(\ddagger)$. 
The two elements $0,1:\ast\to\Omega$ of $\Omega$ can be recovered from $(\ddagger)$ as those corresponding to $\emptyset,\ast\in\text{Subset}(\ast)$.
