Translation:I will not get the newspaper in the shop today but I will get the book tomorrow.
They're called the dependent and independent forms of the verb. They're the same lexeme - FAIGH - in the future tense, but they're used in different situations. Generally, the dependent form, faighidh, is used with the verbal particles, and the independent form gheobhaidh is used without them. There are, of course, exceptions, such as the direct relative clause particle, which uses the independent form.
There are other verbs that have dependent and independent forms, such as bí in the present non-habitual: tá and fuil (ní fhuil is written as níl), and bí, déan, téigh, feic in the past tense.
A simpler explanation is the verb faigh is irregular. This means in the future tense when you say you WILL get something you say gheobhaidh, however when you say you WON'T get something you say ní bhfaighidh