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.