A functor $F\colon C \to D$ is essentially surjective, or essentially surjective on objects (sometimes abbreviated to eso), if it is surjective on objects “up to isomorphism”.
$F\colon C \to D$ is essentially surjective if for every object $y$ of $D$, there exists an object $x$ of $C$ and an isomorphism $F(x) \cong y$ in $D$.
A functor between discrete categories (or, more generally, skeletal categories) is essentially surjective iff it is a surjective function between the classes of objects.
Any bijective-on-objects functor is essentially surjective.
A composition of any two essentially surjective functors is essentially surjective.
If $g f$ is essentially surjective, then $g$ is essentially surjective.
An essentially surjective functor is additionally fully faithful precisely when it is an equivalence of categories.
The inclusion functor of a subcategory is essentially surjective preciesely when the subcategory is essentially wide.
Strengthening the last example, there is an orthogonal factorization system (in the up-to-isomorphism strict sense) on $Cat$, in which eso functors are the left class and fully faithful functors are the right class.
This is an “up-to-isomorphism” version of the bo-ff factorization system, which is a 1-categorical orthogonal factorization system on $Cat$ in which the left class consists of bijective-on-objects functors. Thus essentially surjective is a non-evil version of “bijective on objects”, i.e. the version which views $Cat$ as a bicategory.
In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one. Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.
In any 2-category there is a notion of eso morphism which generalizes the essentially surjective functors in Cat. In a regular 2-category, these form a factorization system in a 2-category together with the ff morphisms.