A finite abelian group has a faithful irreducible representation if and only if it is cyclic. The case of finite groups was solved by Gaschütz in

W. Gaschütz, *Endliche Gruppen mit treuen absolut-irreduziblen Darstellungen.* Math. Nach. 12 (1954)

From Mathematical Reviews:

"In the present formulation the author calls the direct product $$ S= M_1\times M_2\times\cdots\times M_t $$ of the minimal normal subgroups $M_i$ of $G$ the `base' of $ G$, and writes $S=A\times H$, where $A$ is abelian and $H$ contains no normal abelian subgroup. The condition is as follows: a finite group $G$ has a faithful irreducible representation in an algebraically closed field of characteristic zero if and only if the base $S$ (or alternatively, $A$) of $G$ is generated by a single class of conjugates in $G$. The proof is based on an elegant application of the exclusion principle." 

Maybe you also want to look at

Bekka, Bachir; de la Harpe, Pierre, *Irreducibly represented groups.*  Comment. Math. Helv.  83  (2008),  no. 4, 847–868

where the case of infinite groups and unitary representations on Hilbert spaces was studied. One of the main results of the paper is the following:

> **Theorem:** A countable group $G$ is irreducibly represented, if one of the following conditions hold:
>
> 1. $G$ is torsion-free
> 2. $G$ is icc; this means the all non-trivial conjugacy classes are infinite,
> 3. $G$ has a faithful primitive action on an infinite set.