Börnin læra sanngirni þegar þau deila kökunni og bíða rólega eftir röðinni sinni.

Börn means children.
Börnin means the children.

Icelandic usually adds the definite article to the end of the noun instead of using a separate word like the. So:

• börn = children
• börnin = the children

That -in ending is the definite article here.

Because börn is a neuter plural noun in Icelandic, and pronouns normally agree with the noun’s grammatical gender, not natural gender.

So even though the children may be boys, girls, or mixed, the noun börn is grammatically neuter plural, so the pronoun is:

• þau = they (for a neuter plural noun)

This is very common in Icelandic.

They are finite present-tense verbs, not infinitives.

This can be confusing because for many verbs, the 3rd person plural present looks exactly like the infinitive:

• að læra = to learn

• þau læra = they learn

• að deila = to share

• þau deila = they share

• að bíða = to wait
• þau bíða = they wait

So in this sentence, these are real present-tense verb forms. They are not infinitives.

Þegar means when here and introduces a subordinate clause:

• Börnin læra sanngirni = The children learn fairness
• þegar þau deila kökunni og bíða... = when they share the cake and wait...

So the whole second part explains when or under what circumstances the children learn fairness.

Sanngirni is an abstract noun meaning fairness. Abstract nouns are often used without the definite article in Icelandic, just as in English we usually say fairness, not the fairness.

It is also the object of læra:

• læra sanngirni = learn fairness

Its form does not visibly change here, so even though it is the object, it still appears as sanngirni.

Because the verb deila takes the dative when it means share/divide something.

So the noun kaka appears in the dative definite form:

• kakan = the cake (nominative)
• kökuna = the cake (accusative)
• kökunni = the cake (dative)

Since deila governs the dative, the sentence uses kökunni.

This is something you largely have to learn with the verb:

• deila e-u = share something

Because Icelandic, like English, does not need to repeat the subject when two verbs share the same subject.

So:

• þau deila kökunni og bíða rólega...

means:

• they share the cake and wait calmly...

The subject þau applies to both deila and bíða.

Because rólega is an adverb, not an adjective.

It describes how they wait:

• bíða rólega = wait calmly

The adjective is:

• rólegur / róleg / rólegt = calm

The adverb is:

• rólega = calmly

So here it modifies the verb bíða, not the noun þau.

This is part of the expression bíða eftir = wait for.

So:

• bíða eftir e-u = wait for something

This construction takes the dative, which is why we get:

• röðinni = the turn / the order / the place in line (dative definite)

The noun röð can mean things like order, sequence, line, or turn. In this sentence, röðinni is best understood as the turn.

So:

• bíða eftir röðinni = wait for the turn / wait for one’s turn

Because that is a normal Icelandic pattern.

In English, we usually say:

• my book
• their turn

But Icelandic often uses a definite noun + possessive:

• bókin mín = my book
• húsið okkar = our house
• röðinni sinni = their own turn

So even though English does not say the turn their, Icelandic commonly does something structurally similar.

Because sinn is the reflexive possessive. It refers back to the subject of the clause.

Here, the subject of the þegar clause is þau, so sinni means their own:

• þau ... bíða eftir röðinni sinni = they wait for their own turn

If you used þeirra, it would usually be non-reflexive and could suggest someone else’s turn, or at least not point back as neatly to the subject.

So:

• sinn = one’s own / his own / her own / their own, referring back to the clause subject
• þeirra = their, but not reflexive

Because sinn agrees with the thing possessed, not with the possessor.

The possessed noun here is röðinni, which is:

• feminine
• singular
• dative

So the possessive must match that noun, giving:

• sinni

Even though þau is plural, each child is waiting for his or her own turn, and the word turn is singular.

Yes, a little.

In the subordinate clause introduced by þegar, Icelandic uses normal subordinate-clause order:

• þegar þau deila kökunni og bíða...

That is, subject + verb.

This is useful to notice because Icelandic main clauses often follow V2 word order, but subordinate clauses introduced by words like þegar do not work the same way. So this word order is exactly what you would expect in a subordinate clause.