You are probably looking for the complex representations (is that right?). Firstly I claim any finite-dimensional simple rep is one-dimensional. Let $G=\langle g\rangle$ be infinite cyclic, this will stand in place of $\mathbb{Z}$. If $g$ acts on some nonzero finite-dimensional complex vector space $V$ by a linear map $\rho(g)$ then this linear map has an eigenvector, which spans a one-dimensional subrep. If the rep was irreducible, this subrep must be all of $V$, so $\dim V=1$.
Now note that if $\lambda \in \mathbb{C}^*$ (I mean the non-zero complex numbers) then $g \mapsto \lambda$ induces a homomorphism $\rho_\lambda: G \to \mathsf{GL}_1(\mathbb{C})$, i.e. a representation of $G$. Clearly any one-dimensional representation arises this way. Furthermore if $\lambda \neq \mu$ then the representations afforded by $\rho_\lambda$ and $\rho_\mu$ are not isomorphic: not much conjugacy happens in $\mathsf{GL}_1$ !
In conclusion, the simple representations are in natural bijection with $\mathbb{C}^*$: this is called the character group.