The term representation is used for a variety of different types of group homomorphisms.
The first type in the LMFDB are Dirichlet characters. These are homomorphisms $$ \chi:\modstar\Z n \to \C^\times$$ for some $n\in\Z^+$. Typically, they are thought of as functions on $\Z$ via the projection $\Z\to\Z/n\Z$ and extension by $0$.
The LMFDB also contains Artin representations. These are continuous homomorphisms $$ \rho:\Gal(\overline\Q/\Q)\to\GL_n(\C)$$ for some positive integer $n$. Such homomorphisms always factor through the Galois group of some number field, and are complex representations of that Galois group.
A Dirichlet character can be viewed as special type of Artin representation. If $n\in\Z^+$, then $\Gal(\Q(\zeta_n)/\Q)$ can be identified in a natural way with $\modstar\Z n$, and $\C^\times\cong \GL_1(\C)$.
In both cases, the representations can be used to factor Dedekind $\zeta$-functions as products of L-functions.