The short version is that nouns change their shape depending on what role they play in a sentence.
In the sentence "The boy goes to the store with his mother to buy a book" you have a subject who does the action (boy, nominative case) , a direct object that is the thing acted upon (book, accusative case), and two objects that are objects of prepositional phrases (to the store and with the mother).
Different languages treat these differently; for example, English only distinguishes between pronouns (I/me, he/him, she/her, etc.) and other nouns do not change their shape depending on where they are in the sentence, while Russian identifies six different "roles".
If you want a much longer version try here.
I find this site to be pretty helpful and it is also free. I haven't gone deep into the lessons about cases yet but the other lessons I've done were informative. Hope this helps! http://www.russianforfree.com/lessons-russian-language-contents.php
It needs some context, but the general meaning is whether or if the person being asked is aware that she wrote and published a book, if the person has some knowledge of what the book is about, and whether the person may or may not have read it.
There are many aspects about the book which the person may or may not know: how successful it has been, how well-known, what effect it has had, whether it is regarded as a good book or not, etc.
"Knowing" a book implies some unspecified amount of familiarity with it, but may be as simple as only having heard it was published. The person could respond, "Oh, I heard something about that. Is it selling a lot of copies? How have the critics reviewed it?"
Alternatively, the answer could be, "Oh, yes, I've read it twice and I loved it! What a great novel - her second, I believe. She is such a good writer."
Wouldn't it be more proper to state - "Вы знаешь её книгу?"
it seems here, знаете is plural : https://en.wiktionary.org/wiki/%D0%B7%D0%BD%D0%B0%D0%B5%D1%82%D0%B5
whereas, знаешь is for singular: https://en.wiktionary.org/wiki/%D0%B7%D0%BD%D0%B0%D0%B5%D1%88%D1%8C
Why should we знаете here?
All non-nominative forms of он, она, оно, они get an initinal Н after the vast majority of simple prepositions (i.e. «у», «от», «перед», «в», «на», «через» but not «благодаря» thanks to or «согласно» according to):
- У неё есть пицца
- У них с ним контракт
- Мы думаем о нём.
- Я возле них
Note that possessive её, его, их never change and never get the н even if a preposition happens to precede them:
- У неё есть кошка. = She has a cat.
- У её кошки есть котёнок. = Her cat has a kitten.