In the Number Theory for Computing, Song Y. Yan states that Chen used "complicated arguments based on sieve method", when proving what is now called Chen's theorem.
How does this sieve work? Does it has any other applications (i.e. was it applied in other notable proofs or in the theory of computing)?
It seems Chen used what is called "the small sieve".
MR0616545 (82j:10074) Heath-Brown, D. R. Three primes and an almost-prime in arithmetic progression. J. London Math. Soc. (2) 23 (1981), no. 3, 396–414, proved that there exist infinitely many nontrivial arithmetic progressions consisting of three primes and a product of at most two primes, using a method Chen used in his Goldbach paper.
MR0834498 (87g:11079) Harman, Glyn(4-WALC) Diophantine approximation with almost-primes and sums of two squares. Mathematika 32 (1985), no. 2, 301–310 (1986) used Chen's sieve to prove that irrationals have good rational approximations where both numerator and denominator are a sum of two squares.
I'm sure there are many other applications that I didn't find. As to how it works, I believe that's a very technical question --- in any event, it is beyond my competence. There are several books about sieve methods that you could consult.