The localization construction is extremely useful in algebraic geometry.
But this object seems for me very natural (of course, that's maybe only a mistake of my immature mind) for commutative rings itself as a way to describe the properties of rings like $R[x]/\langle ax-1\rangle$ and some more complicated commutative $R$-algebras.
As an example of straightforward applications of the concept in ring theory I could remember at least the least tricky way to prove the description of a nilradical:
For a commutative ring $R$ the following holds: $$\sqrt{\langle 0\rangle}=\bigcap_{I\in \operatorname{Spec} R} I.$$
But are there any more applications of a localization in ring theory. Maybe ones that don't use other constructions?
A very tangible use in basic algebraic number theory is to localize "at" a prime to study primes lying over it... because a Dedekind ring with a single prime (as is the localized version) is necessarily a principal ideal domain, and many arguments become (thereafter) essentially elementary-intuitive. :)