An $n$-group is a group object internal to $(n-1)$-groupoids.
If it is deloopable, an $n$-group $G$ is the hom-object $G = Aut_{\mathbf{B}G}({*})$ of an n-groupoid $\mathbf{B}G$ with a single object ${*}$.
If $\mathbf{B}G$ is a strict n-groupoid, then the corresponding $n$-group is called a strict $n$-group. Strict $n$-groups are equivalent to crossed complexes of groups, of length $n$.
Under the homotopy hypothesis $n$-groups correspond to pointed connected homotopy n-types.
A $1$-group is simply a group; see also 2-group and 3-group.
For $n \lt 1$, there is a single $n$-group, the point.
For arbitrary $n$, there is a circle n-group.
In string theory, we have the string 2-group and the fivebrane 6-group.
See also
The homotopy theory of k-tuply groupal n-groupoids is discussed in