Learn Hardcore Icelandic: Foreldrarnir eru stoltir af börnunum sínum þegar þau tala íslensku við ferðamenn. - The parents are proud of their children when they speak Icelandic with tourists.

Foreldrar means parents (plural). Icelandic usually attaches the definite article to the end of the noun instead of putting a separate word like the in front.

foreldrar = parents
foreldrarnir = the parents

The -nir ending is the masculine plural definite article attached to a noun that ends in -ar in the nominative plural. So:

kennari → kennarar → kennararnir (teacher → teachers → the teachers)
foreldri → foreldrar → foreldrarnir (parent → parents → the parents)

So foreldrarnir simply means the parents as the subject of the sentence.

Why is it eru stoltir and not eru stolt or eru stoltur?

The adjective has to agree with the noun it describes in gender, number, and case.

Subject: foreldrarnir = masculine plural, nominative.
Adjective: stoltur = proud (masculine singular, nominative).

To match masculine plural nominative, stoltur changes to stoltir:

stoltur (m.sg.nom) – one male is proud
stolt (n.sg.nom) – one neuter thing is proud
stoltar (f.pl.nom) – several feminine things/people are proud
stoltir (m./mixed pl.nom) – several males / mixed group are proud

Because foreldrarnir is a grammatically masculine plural noun, we must use stoltir:
Foreldrarnir eru stoltir… = The parents are proud…

Why is it af börnunum sínum and not something like af börnunum þeirra?

Two things are going on here:

The preposition af
af (of, from, about) normally takes the dative case.
The noun börn (children) in dative plural definite becomes börnunum:
- Nom. pl.: börn
- Dat. pl. indef.: börnum
- Dat. pl. def.: börnunum (the children in dative)
So af börnunum = of / about the children.
The possessive sínum
sínum is the dative plural of the reflexive possessive sinn and means their own (referring back to the subject of the sentence).
- Foreldrarnir … af börnunum sínum =
  The parents … (are proud) of their own children.
If you used þeirra instead of sínum, it would usually suggest someone else’s children, not the subject’s:
- Foreldrarnir eru stoltir af börnunum þeirra.
  More likely: The parents are proud of *their children (those other people’s children).*

So af börnunum sínum clearly says: of *their own children* (the children belonging to those same parents).

What exactly is the difference between sínum and þeirra for “their”?

Both can be translated as their, but they work differently.

sínum = reflexive possessive; it always refers back to the subject of the same clause.
þeirra = non‑reflexive possessive; it usually refers to somebody else, not the subject.

In this sentence:

Subject: Foreldrarnir (the parents)
börnunum sínum = their own children (the children of those parents).

Compare:

Foreldrarnir eru stoltir af börnunum sínum.
→ The parents are proud of their own children.
Foreldrarnir eru stoltir af börnunum þeirra kennara.
→ The parents are proud of the children of those teachers
(here þeirra belongs to kennara, not to foreldrarnir).

So sinn/sína/sínum is for “one’s own (subject’s)”, while þeirra is a general “their” that does not necessarily point back to the subject.

Why do we use þau for “they” when referring to börn (children), and not þeir or þær?

Icelandic third‑person plural pronouns agree with the grammatical gender of the noun they refer to, not necessarily the natural gender.

börn (children) is neuter plural.
The neuter plural pronoun is þau.

So:

þeir = they (masculine / mixed, referring to masculine plural nouns)
þær = they (feminine plural)
þau = they (neuter plural)

Since börn is neuter plural, you must use þau:

Foreldrarnir … þegar þau tala…
= The parents … when they (the children) speak…

Why is it þegar þau tala íslensku and not something like þegar þau íslensku tala?

Icelandic word order in a simple subordinate clause like this is usually:

subject – verb – object, just like English.

þegar = when (subordinating conjunction)
þau = they (subject)
tala = speak (verb)
íslensku = Icelandic (object)

So: þegar þau tala íslensku = when they speak Icelandic.

Putting the object before the verb (þegar þau íslensku tala) would sound wrong or at best very marked/poetic. The neutral order is:

Main clause: Foreldrarnir eru stoltir af börnunum sínum
Subordinate clause: þegar þau tala íslensku við ferðamenn

What form is íslensku here, and why doesn’t it have a definite article?

Íslenska is the noun for the Icelandic language (grammatically feminine).

The form íslensku here is accusative singular (weak feminine declension), used as the direct object of tala:

Nom. sg.: íslenska
Acc. sg.: íslensku
(Dat. sg.: íslensku, Gen. sg.: íslensku as well)

The verb tala (to speak) typically takes a language noun in the accusative:

tala ensku – speak English
tala frönsku – speak French
tala íslensku – speak Icelandic

There is also no article because language names in Icelandic are normally used without the definite article in this context, just like in English you say speak Icelandic, not speak the Icelandic.

What does við ferðamenn mean exactly, and why is ferðamenn in that form?

við is a preposition that, among other meanings, can mean to or with when describing talking/interacting with someone.

tala við einhvern = to speak with/to someone

ferðamaður = a tourist, traveler (masc. noun)
Its accusative plural is ferðamenn:

Nom. sg.: ferðamaður (a tourist)
Acc. sg.: ferðamann
Nom. pl.: ferðamenn (tourists)
Acc. pl.: ferðamenn (same form as nom. pl.)

The preposition við takes the accusative case, so in the plural we get við ferðamenn = with/to tourists.

So tala íslensku við ferðamenn = speak Icelandic with tourists or speak Icelandic to tourists.

Why is there no comma before þegar in this sentence?

Modern Icelandic punctuation usually does not put a comma before short subordinate clauses introduced by words like þegar (when), að (that), ef (if), etc., unless there is some special reason (clarity, long inserted phrase, etc.).

So:

Foreldrarnir eru stoltir af börnunum sínum þegar þau tala íslensku við ferðamenn.

is normal and correct without a comma. In English we might choose to add a comma sometimes (“…are proud of their children, when they speak Icelandic…”), but Icelandic is generally more sparing with commas in such cases.

Can the order af börnunum sínum be changed to af sínum börnum? Does it change the meaning?

Yes, you can say af sínum börnum:

Foreldrarnir eru stoltir af börnunum sínum.
Foreldrarnir eru stoltir af sínum börnum.

Both are grammatically correct and both mean “The parents are proud of their own children.”

The differences are subtle:

af börnunum sínum
– Slightly more common and neutral here; the definiteness (börnunum = “the children”) is attached to the noun.
af sínum börnum
– Sounds a bit more “generic” (their own children as a group) and slightly shifts the emphasis towards sínum (“their own”), but in everyday use most people would understand them the same way in this context.

For a learner, you can safely treat them as equivalent here, but note that the case and agreement must still match:

börnunum = dative plural definite
sínum = dative plural reflexive, agreeing with börnunum