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
My current understanding of the paper is that it gives a quite natural way of generating a sequence of ideals, where each ideal is a "cone" (in some sense) of ideals generated by some elements that depend only on the previous ideals. It seems that there is a connection with the conjectured uniform bounds on the dimension of the space of cusp forms of weight k and level N (or rather on the number of generators of those ideals).