Supppose that there is a property of a subgroup that is true for all elements of the group. What do we mean to say by "localising that property"?
For instance, the property of a subgroup $H$ of $G$ being normal in $G$, then to localise the property of normality, we would say that set of elements $g \in G$ such that $H^g = H$, which is equivalently $N_G(H)$.