Rúðuþurrkan er biluð.

It means the windshield wiper.

Icelandic very often builds long compound nouns instead of using several separate words. Here:

• rúðu- relates to rúða = pane, window, windshield
• þurrka here means wiper

So the word is literally something like window/windshield wiper.

It is the definite article attached to the noun.

So:

• rúðuþurrka = a windshield wiper
• rúðuþurrkan = the windshield wiper

Unlike English, Icelandic often puts the at the end of the noun instead of using a separate word.

Yes. The noun rúðuþurrka is feminine.

You can also see that from the adjective in the sentence: biluð is the feminine singular form, and it has to match rúðuþurrkan.

Because Icelandic adjectives must agree with the noun in gender, number, and case.

Here, rúðuþurrkan is:

• feminine
• singular
• nominative

So the adjective also has to be feminine singular nominative:

• masculine: bilaður
• feminine: biluð
• neuter: bilað

That is why the sentence uses biluð.

Er means is.

It is the present-tense singular form of vera, the verb to be.

So the sentence structure is:

• Rúðuþurrkan = the windshield wiper
• er = is
• biluð = broken / defective

In this sentence, it functions as an adjective meaning broken, defective, or out of order.

It is related to the verb bila = to break down / malfunction.

A useful contrast is:

• Rúðuþurrkan er biluð. = The windshield wiper is broken.
  (current state)
• Rúðuþurrkan bilaði. = The windshield wiper broke down.
  (past event)

Usually it means not working properly, malfunctioning, or defective.

So in this sentence, the wiper might be:

• mechanically faulty
• not moving
• not responding

It does not have to mean that it is visibly snapped or shattered.

It is in the nominative singular.

That makes sense because it is the subject of the sentence. The adjective biluð matches it in the same grammatical pattern.

These are two important Icelandic letters:

• þ is like the th in think
• ð is like the th in this

So in this sentence:

• þ in þurrkan has the think sound
• ð in rúðu and biluð has the this sound, though word-final ð is often very weak in actual speech

Yes, but the meaning changes.

• Rúðuþurrka er biluð. = A windshield wiper is broken.
• Rúðuþurrkan er biluð. = The windshield wiper is broken.

In a real situation, if you are talking about the specific wiper on a particular car, the definite form rúðuþurrkan is usually the natural choice.

You would use the plural:

Rúðuþurrkurnar eru bilaðar.

Changes to notice:

• rúðuþurrkan → rúðuþurrkurnar
  singular definite → plural definite
• er → eru
  is → are
• biluð → bilaðar
  adjective agrees with a feminine plural noun