A composition series for a group, $G$, is a subnormal series, (that is, a sequence of subgroups, each a normal subgroup of the next one) such that each factor group $H_{i+1} / H_i$ is a simple group.

An object $X$ of an abelian category has a composition series if there is a chain of subobjects

such that $X_i / X_{i-1}$ is simple for $1\leq i\leq n$.

See at length of an object for more