An $n$-hypergroupoid is a model for an n-groupoid: it is an Kan complex that is like the nerve of a groupoid ($n= 1$), bigroupoid ($n = 2$) etc.
An $n$-hypergroupoid is a Kan complex $K$ in which the horn-fillers are unique in dimension greater than $n$:
(The lower dimensional horn fillers of course also exist, but are not in general unique.)
This is due to (Duskin 79, Glenn 82), however their definition does not ask $K$ has lower dimensional horn fillers. In Beke 04 these are called exact $n$-types instead. For a review on the definition see (Pridham 09, section 2).
Equivalently, this are those Kan complexes which are $(n+1)$-coskeletal and such that the $(n+1)$-horns and $(n+2)$-horns have unique fillers.
2-hypergroupoids are precisely the Duskin nerves of bigroupoids.
The term hypergroupoid is due to Duskin
and his student, Paul Glenn:
The term exact $n$-type is used in
Presentation of higher stacks (higher geometric stacks) by hypergroupoid objects is in