In 1985, Ken Ribet proved a "level-lowering" theorem in the theory of mod p modular forms, valid for forms of level Gamma_0(N) and primes p>2. As a consequence of this result he was able to prove that the Taniyama-Shimura conjecture implied Fermat's Last Theorem. Wiles also used this level-lowering theorem in his initial proof of the semi-stable Taniyama-Shimura conjecture.
Taylor's approach to settling infinitely many new cases of a conjecture of Artin was based on these ideas of Wiles and Ribet, but unfortunately he needed to work with p=2, where Ribet's work, as it stood, did not apply. I will explain Taylor's approach, how one proves level-lowering for p=2, and what the consequences are for this conjecture of Artin.