Sieve Methods
Selberg's sieve, Large sieve, Vaughan's identity, Bombieri-Vinogradoff theorem
Some notes.
Zhang's method
Background
H.Davenport, H.L. Montgomery `Multiplicative number theory', Graduate Texts in Mathematics, 74, Springer, 2000.
Primes in Arithmetic Progression in
H.Iwaniec, E.Kowalski 'Analytic Number Theory', Colloquium Publications 53, AMS, 2004.
GPY and Tao-Maynard methods
Lecture notes-GPY
Lecture note-M_T
Papers
Primes in Tuples
Small Gaps Between Primes
Or Almost Primes
Smaller Gaps