"This book is mine, his is at school."
Translation:Bu kitap benim, onunki okulda.
You only use 'kitabı' when it is both a particular book and also the object of a verb.
So 'Bu kitabı okuyorum' is correct because it's a particular book (this one) AND it's the object of the verb 'okumak' (it's the thing that's being read).
But 'kitap' is used when neither or only one of those conditions is met. For example 'Bir kitap okuyorum.' (It's the object of the verb, but it's just 'a' book, not a particular one) or 'Bu kitap büyük.' (it's a specific book, but not the object of a verb).
This is the accusative case - have a look at the module on it for lots of examples.
Sometimes adjectives do go at the end of the sentence, for example 'my book is big' 'benim kitabım büyük'.
'Bu benim kitabım' means 'this is my book', which is slightly different. 'Benimki' does mean 'mine', but it replaces the noun. So you could say 'bu benimki', meaning 'this is mine', or 'bu kitap benim' meaning this book is mine.