What academic level would one need to be at to fully understand papers published on the twin prim conjecture? Specifically, what academic level would one need to be at to fully understand Goldston-Pintz-Yildirim's work on twin primes? Undergraduate, Graduate, or PhD?
 A: This is a rather subjective question, and it depends on specialization: I suspect there are "pre-docs" (even pre-"grads") studying number theory who would better understand the paper than some PhD-level mathematicians, in say, group theory. Mathematical maturity, and area (and level) of specialization is of greater importance than the degree one holds. It also depends how deeply you want to understand any published paper, including the paper you refer to (and what you mean by "fully" understand.)
A good look at the references will help indicate what prior knowledge and/or familiarity with such work is assumed, as will the abstract. References will often include work that may be a bit more general, and you can backtrack to that work, to check and see what level of familiarity is assumed to understand that work.  
A: Having an undergraduate degree in math from MIT, I'd have to sit down with this paper for a long time before I could understand it... if ever...
At this level of math I think it's less about what general academic level someone has and more about his or her specific area of expertise.
A: Based on the link you have attached It doesn't seem like you need too much. The first 10 or so pages all made perfect sense to me (then again I've read like every prime related article on Wikipedia) 
For this overall:


*

*Read Up on Primes on Wikipedia and click each link, one by one studying each of the pages, each time you come across a word or a term that you don't understand google it, or copy and paste it into Wolfram Alpha and learn it.

*Buy yourself a basic number theory book (ex: Art of Problem Solving Introduction to Number Theory) to get yourself started on understanding how a lot of this is done.

*Read up on some standard High School AP Statistics, and read up some mathematical notation ex: Infinite Product Notation, Infinite Sums, Inclusion - Exclusion...
Thats all the math you'll need to understand the paper. Now you'll just have to move through it slowly, reading each line one by one, and if ever you find a line that makes no sense look it up. 
GOOD LUCK!
