"Ivan is reading our book."
Translation:Иван читает нашу книгу.
Наш 'our' is an adjective-like pronoun, it works like an adjective: it changes its forms to match the gender, number and case of the noun it modifies. Here, «наш» modifies «кни́га» so it matches with its case: in accusative it's «на́шу кни́гу», in genitive it's «на́шей кни́ги», etc.
Мой 'my', твой 'your' (informal singular), ваш 'your' (formal or plural) work in the same way: «Ива́н чита́ет мою́ кни́гу» 'Ivan is reading my book', «Ива́н чита́ет твою́ кни́гу» 'Ivan is reading your book', «Ива́н чита́ет ва́шу кни́гу» 'Ivan is reading your book'.
Another pronoun that works in the same way is сво́й 'one's own', it's a possessive pronoun used when the possessor is the same as the subject of the sentence: «Ива́н чита́ет свою́ кни́гу» 'Ivan is reading his (=Ivan's) book', «Я чита́ю свою́ кни́гу» 'I'm reading my book', «Вы чита́ете свою́ кни́гу» 'You're reading your book'.
Some pronoun, however, don't follow the same pattern. Notably его 'his', её 'her' and их 'their'. These pronoun don't change their form: «Ива́н чита́ет его́ кни́гу» 'Ivan is reading his (=not Ivan's) book', «Ива́н чита́ет её кни́гу» 'Ivan is reading her book', «Ива́н чита́ет их кни́гу» 'Ivan is reading their book'.
Isnt ваня the same as Ivan? It says to select ALL correct translations but marked me wrong for that one.
Vanya relates to Ivan as Dick does to Richard, Peggy does to Margaret and Billy does to William. Not the same thing at all, but rather a diminiutive form thereof. If they wanted you to use Ваня then they would have asked for it. Иван is always Ivan.