Ég keypti bókina handa henni.

Because bókina is the form that matches both:

• definiteness: the book, not just a book
• case: it is the direct object of keypti

The noun bók is feminine, and its relevant singular forms are:

• bók = a book
• bókin = the book, nominative
• bókina = the book, accusative

After keypti in this sentence, the noun is the object, so Icelandic uses the accusative: bókina.

In Icelandic, the definite article is usually attached to the end of the noun instead of being a separate word.

So instead of a separate word like English the, Icelandic often says:

• bók = book / a book
• bókin = the book
• bókina = the book in the accusative

So the -in / -ina part is doing the job of the, while also reflecting case.

Because handa takes the dative case, and the dative form of hún is henni.

The singular forms of hún are:

• hún = she
• hana = her, accusative
• henni = her, dative
• hennar = her, genitive

Since the sentence uses handa, the pronoun after it has to be dative, so henni is correct.

Handa is a preposition meaning something like for, especially in the sense of intended for or for the benefit of someone.

In a sentence like this, it marks the person who is meant to receive the thing:

• Ég keypti bókina handa henni.
• I bought the book for her.

It is very natural when talking about buying, making, or getting something for someone.

Sometimes yes, but the nuance is not exactly the same.

• handa henni strongly suggests intended for her / for her to have
• fyrir hana is broader and can mean for her, but fyrir has many other uses too

With buying a gift or something meant for someone, handa is often especially clear and idiomatic.

So in this sentence, handa henni is a very good choice because it clearly marks her as the intended recipient.

Kaupa means to buy.
Keypti is its 1st person singular past tense form: I bought.

A small stem change happens:

• kaupa = to buy
• ég kaupi = I buy
• ég keypti = I bought

This kind of vowel change is normal in Icelandic verbs, so learners usually just memorize the main forms.

Here is the breakdown:

• Ég = subject, I
• keypti = verb, past tense, bought
• bókina = direct object, accusative definite, the book
• handa henni = prepositional phrase, for her

So the structure is basically:

subject + verb + object + prepositional phrase

Not completely. Icelandic word order is more flexible than English because case endings help show each word’s role.

The sentence as given is a very natural neutral order:

• Ég keypti bókina handa henni.

But Icelandic can move things around for emphasis more easily than English. For example, the handa henni part could be placed differently in some contexts.

Still, for a learner, this sentence is a very good standard pattern to follow.

Because many Icelandic verbs take a direct object in the accusative, and kaupa is one of them.

So:

• ég keypti bókina
• I bought the book

The thing bought is the direct object, and that is why bókin becomes bókina here.

A rough English-friendly approximation is:

yeg kyap-ti boh-ki-na han-da hen-ni

A few helpful points:

• Ég begins with a y-like sound
• ey in keypti sounds roughly like ay
• ó is a long o
• ll words are not involved here, so this sentence is relatively straightforward
• Icelandic usually stresses the first syllable of each word

So the main stresses are on:

• ÉG
• KEYP-ti
• BÓ-ki-na
• HAN-da
• HEN-ni

Yes, very much. Icelandic pronouns change form often, and these forms appear constantly.

For hún, the key singular forms are:

• hún = she
• hana = her, accusative
• henni = her, dative
• hennar = her, genitive

In this sentence, learning henni is especially useful because many prepositions require the dative.