Hún sker gulrót og gúrku smátt og setur þær í litla skál.

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

So:

• gulrót = a carrot or just carrot
• gúrku = a cucumber or just cucumber
• litla skál = a small bowl

If Icelandic wants to say the, it usually adds the definite article to the end of the noun:

• gulrótin = the carrot
• gúrkan = the cucumber
• skálin = the bowl

Because gúrku is the accusative singular form of gúrka.

The verb skera takes a direct object in the accusative, so the thing being cut changes case:

• nominative: gúrka
• accusative: gúrku

So in Hún sker ... gúrku, the cucumber is the object, and that is why the form is gúrku.

Because not every noun shows a visible change in the accusative.

Gulrót is also a direct object here, so it is accusative too. But for this noun, the accusative singular happens to look the same as the dictionary form:

• nominative: gulrót
• accusative: gulrót

So both gulrót and gúrku are objects of sker, but only gúrka shows a different accusative ending.

Here smátt means something like finely, into small pieces, or small in the sense of how the cutting is done.

It is functioning adverbially: it describes how she cuts the vegetables.

A useful thing to know is that Icelandic often uses the neuter singular form of an adjective in an adverb-like way. So smátt is connected to the idea of small, but here it means:

• cut finely
• cut into small pieces

So sker ... smátt is very natural Icelandic.

That is a normal Icelandic word order.

In English, learners often expect something like she finely cuts or she cuts finely. Icelandic is often comfortable placing this kind of adverbial expression after the object:

• sker gulrót og gúrku smátt

This still means that smátt describes the cutting, not the nouns themselves. In other words, it means she cuts the carrot and cucumber finely / into small pieces.

Þær refers back to gulrót og gúrku.

Those two nouns are both feminine:

• gulrót = feminine
• gúrka = feminine

Together they form a plural group, so Icelandic uses the feminine plural pronoun:

• þær = they / them for an all-feminine plural group

In this sentence it means them.

Because the pronoun agrees with the gender of the nouns it refers to.

Since both nouns are feminine, the plural pronoun is feminine too:

• þær = feminine plural
• þeir = masculine plural
• þau = neuter plural, often used for mixed groups

So þær is correct because both gulrót and gúrka are feminine nouns.

Also, even though each noun is singular by itself, together they make a plural idea, so a plural pronoun is needed.

Because í can take either accusative or dative, depending on meaning.

• accusative = movement into
• dative = location in

Here she is putting the vegetables into the bowl, so there is movement. That is why Icelandic uses the accusative:

• í litla skál = into a small bowl

If you were describing location instead, you would use the dative:

• í lítilli skál = in a small bowl

So this is a very common Icelandic pattern:

• movement: í
  • accusative
• location: í
  • dative

Because the adjective has to agree with skál in gender, number, and case.

The dictionary form is lítil for feminine singular nominative, but here skál is in the accusative singular after í with movement, so the adjective changes form too:

• nominative: lítil skál
• accusative: litla skál

So litla is simply the correct agreeing form in this sentence.

Because the subject is the same in both parts of the sentence.

Icelandic, like English, often avoids repeating the subject when two verbs are joined by og and both have the same subject:

• Hún sker ... og setur ...

This works just like English:

• She cuts ... and puts ...

Repeating hún is possible in some contexts, but it is not necessary here.

The basic meaning of skera is to cut.

But in a sentence like this, English may translate it more naturally as:

• cuts finely
• chops
• cuts into small pieces

That is because of smátt, which tells you the cutting is done into small pieces. So sker is literally cuts, but the best English translation may be chops or cuts finely, depending on style.