My initial answer was, it turns out, weak. So I'm going to add in some of the observations made in comments, and turn this answer into community wiki, so as not to get credit.
If G is principle, then the factors of its upper central series, lower central series and derived series are cyclic. There are many other immediate consequences of definition: If G is principal, then so are its quotients and its characteristic subgroups. Such a group has the max-n property. If it is hypercentral, then it is nilpotent. If solvable, then super-solvable.
For infinite groups, the observation of the previous bullet implies that Baumslag and Blackburn's paper [``Groups with Cyclic Upper Central Factors''][1] is relevant.
For finite $p$-groups, observe that any principal p-group P is cyclic as the Frattini quotient $P/[P,P]∗P^p$ is one dimensional (again, this follows from the first bullet point). This implies, in particular, that a finite principal nilpotent group is cyclic.
As for finite principal solvable groups, well, things are less clear. Note that $S_3$ is principal.
If $G$ is a finite $p$-group such that its proper normal subgroups are principal ($G$ itself is not principal), then $G$ has maximal class. The converse is also true.
If $G$ is finite nilpotent group such that its proper normal subgroups are principal, then $G$ is cyclic or it is a $p$-group of maximal class. [1]: http://plms.oxfordjournals.org/content/s3-10/1/531.short
