Ég kaupi gjöf fyrir ömmu mína.

• Ég = I
• kaupi = buy / am buying
• gjöf = gift
• fyrir = for
• ömmu = grandmother / grandma in the form required here
• mína = my in the form required here

So the structure is basically:

I + buy + gift + for + my grandmother

Kaupi is the 1st person singular present tense form of the verb að kaupa (to buy).

So:

• að kaupa = to buy
• ég kaupi = I buy / I am buying

This is similar to how English changes to be into I am, he is, etc. Icelandic verbs also change depending on the subject.

It can mean either, depending on context.

In Icelandic, the simple present often covers both:

• I buy
• I am buying

So Ég kaupi gjöf fyrir ömmu mína can mean:

• I buy a gift for my grandmother
• I am buying a gift for my grandmother

The surrounding context usually makes it clear.

Because fyrir takes the accusative case when it means for in a sentence like this, and amma changes form in the accusative.

The noun is:

• amma = grandmother / grandma

But in this sentence, after fyrir, it becomes:

• ömmu

So:

• amma = basic dictionary form
• ömmu = accusative form used here

This kind of noun change is very common in Icelandic.

Because the possessive word mín (my) has to match the noun it describes in gender, number, and case.

Here it describes ömmu, which is:

• feminine
• singular
• accusative

So the correct form is mína.

You can think of it like this:

• mín = a base form you may see in dictionaries
• mína = the form that agrees with ömmu

This agreement is an important feature of Icelandic grammar.

Because Icelandic marks grammar in more than one place.

• The noun amma changes to ömmu
• The possessive mín changes to mína

Both are showing the same grammatical role: they belong together as for my grandmother, and that phrase is in the accusative after fyrir.

English usually does not do this much, so it can feel unusual at first.

Icelandic does not have an independent indefinite article like English a/an.

So:

• gjöf can mean gift or a gift

The sentence does not need a separate word for a.

If Icelandic wants to express the gift, it usually adds a definite ending to the noun instead.

For example:

• gjöf = a gift / gift
• gjöfin = the gift

Because gjöf here is indefinite: a gift, not the gift.

• gjöf = a gift
• gjöfin / gjöfina = the gift (different case forms depending on the sentence)

So Ég kaupi gjöf fyrir ömmu mína means I am buying a gift for my grandmother, not specifically the gift.

Gjöf is in the accusative singular here, because it is the direct object of kaupi.

You are buying what?

• gjöf

In Icelandic, direct objects are often in the accusative.

So the sentence has:

• Ég = subject
• kaupi = verb
• gjöf = direct object in the accusative
• fyrir ömmu mína = prepositional phrase

Fyrir can take different cases depending on meaning, which is something Icelandic learners often have to memorize.

In this sentence, fyrir means for, and it takes the accusative:

• fyrir ömmu mína = for my grandmother

In other contexts, fyrir can also take the dative. So it is one of those prepositions whose case depends on usage and meaning.

Yes, Icelandic word order is fairly flexible, although the neutral order here is very natural:

• Ég kaupi gjöf fyrir ömmu mína.

You may also see other orders for emphasis, for example:

• Gjöf kaupi ég fyrir ömmu mína.

That sounds more marked, like emphasizing gift.

For a beginner, it is best to learn the neutral pattern first:

• Subject + verb + object + prepositional phrase

No. It is capitalized here only because it is the first word of the sentence.

The pronoun ég means I, but unlike English I, it is not always capitalized in the middle of a sentence.

So:

• At the beginning: Ég
• In the middle: ég

That is a very common thing for English speakers to notice.

A rough learner-friendly guide is:

• gjöf ≈ something like gyev or gyœv

A few helpful points:

• gj at the beginning sounds somewhat like gy
• ö is a rounded vowel that English does not have exactly
• The final f is pronounced like v in this word

So it does not sound like an English f at the end.

A rough approximation is:

• ömmu ≈ UHM-mu with a rounded vowel at the start
• mína ≈ MEE-na

A few notes:

• ö is not the same as English o
• mm is a long doubled consonant
• í is like ee in see
• Stress in Icelandic usually falls on the first syllable

So:

• ÖMM-u MÍ-na

Yes, but not in this exact sentence.

• amma mín is a valid combination meaning my grandmother in the nominative
• in this sentence, after fyrir, you need the accusative
• so it becomes ömmu mína

So:

• Amma mín er hér. = My grandmother is here.
• Ég kaupi gjöf fyrir ömmu mína. = I buy a gift for my grandmother.

This is a good example of how Icelandic changes both nouns and possessives depending on case.