If X is a type of structure that can be defined in a category, higher category, or category with some sort of structure, then the walking X is an informal term for the free category (resp. higher category, category with suitable structure) containing an X.
More precisely, if $StructCat$ denotes some (higher) category of categories with an appropriate type of structure, then the walking X is an object $[X] \in StructCat$ together with a natural equivalence
between the hom-set/category/space from $[X]$ to $C$, for any $C\in StructCat$, and the set/category/space of all Xs in $C$.
The interval category is the walking arrow.
The augmented/algebraist’s simplex category is the walking monoid (in a monoidal category).
The syntactic category of a theory $T$ in some doctrine $D$ is the “walking $T$-model” (in a $D$-category). In particular, the classifying topos of a geometric theory $T$ is the “walking $T$-model” qua Grothendieck topos (where the morphisms are the left-adjoint parts of geometric morphisms).