Frobenius groups have trivial center. This is helpful for examplesnon-examples, e.g., to see that the$p$-groups, or the quaternion group is not a Frobenius (and all nilpotent groups). Further examples are: non-abelian groups of order $pq$, the group $A_4$. Further non-examples: the groups $S_n$ for all $n\ge 4$.
In addition to the other answers, here is a nice discussion on Frobenius groups by Terence Tao in his tag archive: http://terrytao.wordpress.com/tag/frobenius-groups/.