I am currently trying to decipher Mazur's Eisenstein ideal paper (not a comment about his clarity, rather about my current abilities). One of the reasons I am doing that is that many people told me that the paper was somehow revolutionary and introduced a new method into number theory. 
Could you explain exactly what subsequent developments did the paper bring, what ideas in the paper were considered more-or-less original (at the time it was published), and exactly what difficulties did these ideas resolve that people failed to resolve before the paper was published (if any)?
The Mazur's paper "Ideals in \(p\)-adic number fields and arithmetic of elliptic curves" (Annals of Mathematics, 1972) is indeed seminal in the development of \(p\)-adic number theory and its applications to arithmetic geometry. One can divide the impact of the paper into several areas:

1. New constructions and classification of \(p\)-adic Lie alge
In particular, you mentioned that "the paper introduced a number of fundamental concepts in the theory of ideal classes in number fields" -- I was wondering if you could give some examples of these concepts and explain their significance.
I can only add one technical comment: I was under the impression that the paper was actually joint with Serre. I see that you acknowledge Serre's help later on, but somehow this fact seems not to be explicitly indicated at the beginning of the paper.
Thank