What is the necessary and sufficient condition for an integral domain to have gcd for every pair of elements and why?
For a silly condition: every pair of elements in a domain $D$ has a gcd if and only if every pair of elements in $D$ has an lcm.
More seriously, a good survey is:
GCD domains, Gauss' Lemma, and content of polynomials by D.D. Anderson. In Non-Noetherian commutative ring theory, pages 1-13. Math. Appl. 520, Kluwer Acad, Publ., Dordrecht, 2000. MR 1858155 (2002g:13039).
Though, in general, the equivalent conditions are probably not what you are looking for (for example, one of the equivalent conditions is that a domain $D$ is a GCD-domain if and only if the group of divisibility of $D$ is a complete lattice ordered group).