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)?
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).
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