CW answer.

As already mentioned in a comment, the question was already asked at MathSE. The property in consideration is "every nontrivial quotient is isomorphic to the group itself".

In summary:

• the trivial group satisfies this property
• simple groups satisfy this property
• as mentioned in this answer, finitely generated groups with this property are trivial or simple (indeed, every nontrivial f.g. group has a simple quotient). This more generally holds for groups that are finitely normally generated.
• as mentioned in the same answer the most obvious nontrivial nonsimple examples are Prüfer (aka quasi-cyclic) groups $C_{p^\infty}$ for $p$ prime. These are the only ones among abelian groups.
• nonabelian nonsimple examples are provided in this answer. (As the Prüfer groups, they are locally finite; these ones are direct limits of iterated wreath products of finite alternating groups.)