(1) The referent of the NP be either familiar within the discourse ... --- Why does it have to be 'within the discourse'? Can't it be familiar outside the discourse? I mean, what are the authors saying here? I'm confused. Does it matter whether it is familiar within the discourse or anywhere?
(2) Are the authors saying 'the referent of the NP be either familiar within the discourse to the hearer? (Yes, right?)
Research into the meaning of the English definite article has generally been approached from one of two perspectives, characterizable as 'familiarity' and 'uniqueness.' That is, felicitous use of the definite article has been argued to require that the referent of the NP be either familiar within the discourse or uniquely identifiable to the hearer. The vast majority of uses can be accounted for under either view, since an entity typically must be familiar in a given discourse in order to be identifiable to the hearer.
*NP: noun phrase, i.e. a noun with modifiers, or just a bare noun. (This annotation is added by Sssamy)
('Uniqueness, Familiarity, and the Definite Article in English' by Betty Birner and Gregory Ward)