Let G be a finite group. In general case, given two normal subgroups N and K of G, we need not to have N < K or K<N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume that G has such a property, that is the normal subgroups of G, constitute a chain with respect to inclusion.  For example, simple groups, cyclic groups and symmetric groups satisfy this property. Certainly, if G has such a property, then the normal subgroups are necessarily characteristic. Also we may find out that the center of G must be cyclic. Indeed, every abelian group with this property is cyclic (and vice versa). This also shows that G/G' is cyclic, for, the property in quotient hereditary. 
Please let me know, if these groups are studied before.