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.

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.