A set of generators is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing all the elements in the group. Cyclic groups can be generated as powers of a single generator. Two elements of a dihedral group that do not have the same sign of ordering are generators for the entire group.
The Cayley graph of a group and a subset of elements (excluding the identity element) is connected iff the subset generates the group.