"I need a book."
While your version, being in biernik (accusative) is quite widely used (1880 results in google for the exact phrase, vs 4080 for "potrzebuję książki"), the linguists would say that the only case which can be used with "potrzebować" is dopełniacz (genitive), thus making "Potrzebuję książki" the only really correct option.
Why is it książki and not książky? I'm using the grammar book "Polish Verbs & Essentials of Grammar" (see pg 18) and the declension for the genitive case of książka should be książky because książka is a hard stem feminine noun. Is this noun an exception to the rule? Why/Why not?
Książki is both nominative/accusative plural and genitive singular. In this particular sentence it's genitive singular, since potrzebować is one of those verbs that always takes the genitive instead of accusative.