U genitivu nema knjige, a u dativu dajem knjigu sestri.

They mean in the genitive and in the dative.

Here the sentence is talking about grammar cases, not about physical location. So u genitivu means in the genitive case, and u dativu means in the dative case.

Because after the preposition u meaning in, Croatian normally uses the locative case.

So:

• genitiv → u genitivu
• dativ → u dativu

A useful detail: even though the phrase says in the dative, the word dativu itself is not in the dative here; after u, it is functioning as locative. For these nouns, the locative singular happens to look the same as the dative singular.

Nema is the 3rd person singular form of nemati.

It can mean:

• there is no / there are no
• does not have

In this sentence, nema knjige means something like there is no book.

This use is very common in Croatian.

Because nema normally requires the genitive for the thing that is absent.

So:

• knjiga = nominative singular
• knjige = genitive singular

That is why you get nema knjige.

This is one of the most important genitive patterns in Croatian:

• ima knjigu is not correct for this meaning
• nema knjige is the standard form for there is no book

Because knjigu is the direct object of dajem.

In Croatian, the direct object is usually in the accusative.

So:

• knjiga = nominative singular
• knjigu = accusative singular

In dajem knjigu sestri, the thing being given is the book, so knjiga changes to knjigu.

Because sestri is the recipient — the person to whom something is given.

Croatian uses the dative for that role.

So:

• sestra = nominative singular
• sestri = dative singular

Therefore:

• dajem knjigu sestri = I give the book to my sister / to the sister

Because Croatian has no articles.

English says:

• a book
• the book

Croatian usually just says:

• knjiga
• knjigu
• knjige

The exact meaning depends on context. If needed, Croatian can be more specific with words like:

• jedna knjiga = a/one book
• ta knjiga = that book / the book

But very often nothing extra is needed.

Because a often marks a contrast or a shift of topic.

Here the sentence is comparing two case examples:

• u genitivu ...
• a u dativu ...

So a works like:

• and on the other hand
• whereas
• while

If you used i, it would sound more like simple addition. A is more natural here because the sentence is contrasting two grammatical situations.

Yes, dajem is present tense.

Depending on context, it can mean:

• I give
• I am giving

In this sentence, the exact English choice depends on how the meaning was presented, but the Croatian form itself is normal present tense.

Also, dajem is an imperfective form, which fits a general example sentence well.

Because Croatian endings can be ambiguous.

Knjige can be:

• genitive singular of knjiga
• nominative plural of knjiga

So the same form can mean different things in different contexts.

Here, because it follows nema, it is understood as genitive singular:

• nema knjige = there is no book

Context tells you which meaning is intended.

Yes. Croatian word order is more flexible than English because the case endings show the grammatical roles.

For example, the second clause could also appear as:

• Sestri dajem knjigu
• Knjigu dajem sestri

These versions change the emphasis, but the basic meaning stays the same.

The version in your sentence, dajem knjigu sestri, is a clear and neutral order for a learner example.