Typically used for an element or a subset in a set with additional structure, satisfying certain special core properties.
In group theory
Center of a group: The set of those elements of the group that commute with every element of the group. It forms a subgroup, in fact, a characteristic subgroup.
Primary subject wiki entry: Groupprops:Center
Also located at: Wikipedia:Center (group theory)
In ring theory
Center of a ring: The set of those elements of a (usually unital) ring that commute with every element of the ring. The center of a ring is a subring; for a unital ring, it is a subring containing the multiplicative identity.
Primary subject wiki entry: Noncommutative:Center of a ring