In the abstract Bourbaki set-up, the affine Weyl group is defined to be the semidirect product of an irreducible Weyl group and its coroot lattice. This is naturally a Coxeter group, characterized in terms of its positive semidefinite Coxeter matrix. The basic theory is developed independently of applications in Lie theory, but is directly usable if you start with a connected semisimple algebraic group (over an algebraically closed field) and require its root system to be irreducible of type A, B, etc. Most of the time this causes no trouble. While it is natural to work with a connected reductive group, people often use the expression "affine Weyl group" too loosely in this general context. For example, the standard features of alcove geometry require the irreducibility. Otherwise you have to deal with products of simplexes, etc.
In the Iwahori-Matsumoto (or Bruhat-Tits) setting over local fields, a more intrinsic affine Weyl group occurs directly within the structure of the group itself. But here too one has to be cautious about applying abstract Coxeter group theory or BN-pair theory, as I believe most authors are. In other situations (the classical study of compact Lie groups or the more recent application of affine Weyl groups in modular representation theory starting with Verma) there is usually no difficulty about specializing to the irreducible case. Here the affine Weyl group lives outside the actual group under study. This is the situation I'm most comfortable with. You probably need to specify more precisely the setting in which you really want to study reductive groups, to adapt the Bourbaki language and results.