The notion of $k$-surjective functor is the continuation of the sequence of notions
essentially surjective and full functor
essentially surjective and full and faithful functor
from category theory to an infinite sequence of notions in higher category theory.
Roughly, a functor $F : C \to D$ between ∞-categories $C$ and $D$ is $k$-surjective if for each boundary of a k-morphisms in $C$, each $k$-morphism between the image of that boundary in $D$ is in the image of $F$.
For the moment, this here describes the notion for globular models of $\infty$-categories. See below for the simplicial reformulation.
An $\omega$-functor $f : C \to D$ between $\infty$-categories is 0-surjective if $f_0 : C_0 \to D_0$ is an epimorphism.
For $k \in \mathbb{N}$, $k \geq 1$ the functor is $k$-surjective if the universal morphism
to the pullback $P_k$ in
coming from the commutativity of the square
(which commutes due to the functoriality axioms of $f$) is an epimorphism.
If you interpret $C_k$ and $P_k$ as sets and take ‘epimorphism’ in a strict sense (the sense in Set, a surjection), then you have a strictly $k$-surjective functor. But if you interpret $C_k$ and $P_k$ as $\infty$-categories or $\infty$-groupoids and take ‘epimorphism’ in a weak sense (the homotopy sense from $\infty$-Grpd), then you have an essentially $k$-surjective functor; equivalently, project $C_k$ and $P_k$ to $\omega$-equivalence-classes before testing surjectivity. A functor is essentially $k$-surjective if and only if it is equivalent to some strictly $k$-surjective functor, so essential $k$-surjectivity is the non-evil notion.
For $C$ and $D$ categories we have
An $\omega$-functor $f : C \to D$ is $k$-surjective for $k \in \mathbb{N}$ precisely if it has the right lifting property with respect to the inclusion $\partial G_{k} \to G_k$ of the boundary of the $k$-globe into the $k$-globe.
One recognizes the similarity to situation for geometric definition of higher category. A morphism $f : C \to D$ of simplicial sets is an acyclic fibration with respect to the model structure on simplicial sets if it all diagrams
have a lift
for all $k$, where now $\Delta[k]$ is the $k$-simplex.
With respect to the folk model structure on $\omega$-categories an $\omega$-functor is
an acyclic fibration if it is $k$-surjective for all $k \in \mathbb{N}$;
a weak equivalence if it is essentially $k$-surjective for all $k \in \mathbb{N}$. See also equivalence of categories.
All this has close analogs in other models of higher structures, in particular in the context of simplicial sets: an acyclic fibration in the standard model structure on simplicial sets is a morphism $X \to Y$ for which all diagrams
have a lift
This is precisely in simplicial language the condition formulated above in globular language.
The general idea of $k$-surjectivity is described around definition 4 of
The concrete discussion in the context of strict omega-categories is in
For the analogous discussion for simplicial sets see
and references given there.