Dirichlet characters appear in Dirichlet $L$-functions, in Gauss sums, and in other arithmetical generating functions. They are not exactly group characters, but are extensions by $0$ of such.
Dirichlet characters are multiplicative homomorphisms from $(\mathbb Z/N\mathbb Z)^\times$ to $\mathbb C^\times$, extended by $0$ to all of $\mathbb Z/N\mathbb Z$, and then composed with the quotient map $\mathbb Z\to \mathbb Z/N\mathbb Z$.
