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)?
To answer your question, Mazur's paper "Torsion Points and Complex multiplication" [1] introduced several important ideas into the study of arithmetic algebraic geometry and complex multiplication. The paper was published in 1972 and was groundbreaking for several reasons:

1. Computation of complex multiplication: Mazur introduced a method to compute complex multiplications explicitly using torsion points on elliptic curves. This allowed for the first
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).