The following passage comes from Eisenbud's "Commutative Algebra with a View Towards Algebraic Geometry."
... localization as a general procedure was defined rather late: In the case of integral domains it was described by Grell, a student of Noether, in 1927, and it was not defined for arbitrary commutative rings until the work of Chevalley  and Uzkov , long after the basic ideas of commutative algebra were established. Perhaps this is because interest was focused on finitely generated algebras on the one hand, and power series rings on the other, and neither of these classes of rings are closed under localization. Instead of passing to a localized rings, as we would now, people often used ideal quotients as a substitute.
The idea of ideal quotients is explored in the exercises, so I am willing to be patient on that notion. However these early notions of localization sound interesting to me, and it would be nice if someone here with some knowledge of commutative algebra could summarize the different notions of localization that have existed historically, and maybe discuss the evolution of the idea. If nothing else, a more bird's eye view of the history than what I could find in individual papers may be nice, hunting them down gives me the impression that it will be hard to piece together how the concept evolved from technical publications.