Use with the (group-theory) tag. A group is simple if it has no proper, non-trivial normal subgroups. Equivalently (for finitely generated groups), its only homomorphic images are itself and the trivial group. The classification of finite simple groups is one of the great results of modern mathematics.

A group is simple if it has no proper, non-trivial normal subgroups (a subgroup $H\leq G$ is proper if $G\neq H$, and is non-trivial is $H\neq 1$). Equivalently (for finitely generated groups), its only homomorphic images are itself and the trivial group. The classification of finite simple groups is one of the great results of modern mathematics.

Simple groups can be seen as the "building blocks" of groups. This is explained in this question.