Terminology
The term localization originates in algebraic geometry: if R is a ring of functions defined on some geometric object (algebraic variety) V, and one wants to study this variety "locally" near a point p, then one considers the set S of all functions which are not zero at p and localizes R with respect to S. The resulting ring R* contains only information about the behavior of V near p. Cf. the example given at local ring.
In number theory and algebraic topology, one refers to the behavior of a ring at a number n or away from n. "Away from n" means "in the ring localized by the set of the powers of n" (which is a Z-algebra). If n is a prime number, "at n" means "in the ring localized by the set of the integers which are not multiple of n".
Read more about this topic: Localization Of A Ring