So for $C$ an $\omega$-category, its $\omega$-nerve is the simplicial set whose $k$-simplices are precisely all possible images of the $k$-oriental in $C$:

$N(C)_k := \omega Cat(O[k], C)
\,.$

Where the $\omega$-category itself provided rules for how exactly to compose k-morphisms, its $\omega$-nerve just records all possible ways of how $(k+1)$-morphisms connect pasting diagrams of $k$-morphisms in $C$. This is however precisely the same information.

Characterization of $\omega$-categories by their $\omega$-nerves

Accordingly, omega-nerves may be used to define and identify $\omega$-categories. For instance

the $\omega$-nerve $N(C)$ is a simplicial set in which allhorns have unique fillers precisely if $C$ is a 1-groupoid;

the $\omega$-nerve $N(C)$ is a simplicial set in which all innerhorns have unique fillers precisely if $C$ is an ordinary category;

the $\omega$-nerve $N(C)$ is a simplicial set in which allhorns have any fillers precisely if $C$ is an ∞-groupoid (see Kan complex for more on this);

the $\omega$-nerve $N(C)$ is a simplicial set in which all innerhorns have any fillers precisely if $C$ is an (∞,1)-category.

in full generality, a simplicial set is the $\omega$-nerve of an $\omega$-category if it is a weak complicial set.