Borðið er óhreint eftir morgunmatinn.

Because borðið means the table.

In Icelandic, the definite article the is usually attached to the end of the noun instead of being a separate word.

• borð = table
• borðið = the table

Here, borð is a neuter noun, and -ið is the definite ending for this form.

Yes, in other contexts borðið can also be a form of the verb borða (to eat), specifically the plural/formal imperative eat!

So Icelandic learners often notice this ambiguity:

• borðið = the table
• borðið! = eat! (said to several people, or formally)

In this sentence, it clearly means the table because of the context:

• Borðið er óhreint = The table is dirty

A command like Eat is dirty would make no sense here.

Because the adjective has to agree with borðið.

Since borðið is:

• singular
• neuter
• nominative

the adjective must also be in the neuter singular form:

• óhreinn = dirty, masculine singular
• óhrein = dirty, feminine singular
• óhreint = dirty, neuter singular

So:

• Borðið er óhreint = The table is dirty

This is standard adjective agreement in Icelandic, even when the adjective comes after er (is).

The prefix ó- usually gives a negative meaning, similar to English un- or in-.

So:

• hreinn = clean
• óhreinn = unclean / dirty

In this sentence, óhreint is just the neuter singular form of that adjective.

For the same basic reason as borðið: Icelandic usually adds the to the end of the noun.

• morgunmatur = breakfast
• morgunmatinn = the breakfast

Here, the noun is in a form where the definite ending appears as -inn.

So the sentence literally contains after the breakfast, though in natural English we often just say after breakfast.

Because the preposition eftir takes different cases depending on meaning, and here it means after in a time sense.

With the meaning after, following, it takes the accusative.
That is why we get:

• morgunmatur → dictionary form
• morgunmatinn → accusative definite form

So:

• eftir morgunmatinn = after breakfast

This is a very common pattern in Icelandic.

In this sentence, it means after breakfast in the time sense: the table is dirty following breakfast.

So the idea is:

• breakfast happened
• afterwards, the table is dirty

The phrase does not mean that the table is somehow physically made of breakfast leftovers; it just gives the time/context.

Icelandic often uses the definite form in places where English uses no article.

So even though morgunmatinn literally looks like the breakfast, the natural English translation is usually just:

• after breakfast

This is normal and not something you should always translate word-for-word.

Sometimes you may hear article-less expressions in Icelandic, but in a sentence like this, eftir morgunmatinn is a very natural and standard way to say after breakfast referring to the breakfast meal in question.

For a learner, the safest choice here is:

• eftir morgunmatinn

If you drop the article, the phrase may sound a bit different in style or less like the standard expression in this context.

The basic structure is:

• Borðið = subject
• er = verb
• óhreint = predicate adjective
• eftir morgunmatinn = time phrase / prepositional phrase

So literally:

• The table
  • is
    • dirty
      • after breakfast

This is a very normal Icelandic sentence pattern.

Yes. er is the present tense singular form of vera (to be).

So:

• ég er = I am
• þú ert = you are
• hann/hún/það er = he/she/it is

In this sentence:

• Borðið er óhreint = The table is dirty

A rough learner-friendly pronunciation is:

BOR-dhith er OH-hreint EP-tir MOR-gun-mah-tin

A few notes:

• ð in borðið is a soft sound like th in this
• ó is a long o sound
• ei in hreint sounds somewhat like ay
• ll is not present here, so pronunciation is a bit easier than in many Icelandic sentences

A more careful approximation would vary by speaker, but the main thing is to recognize the forms when you hear them.

It is a very normal and useful word, especially in the sense of not clean.

So:

• hreinn = clean
• óhreinn = dirty / unclean

Depending on context, Icelandic can also use other words for dirty, such as ones meaning filthy, messy, or stained, but óhreinn is a very straightforward everyday choice here.

It is in the nominative singular definite.

That is because it is the subject of the sentence:

• Borðið er óhreint = The table is dirty

A good beginner breakdown is:

• borð = table
• borðið = the table
• case here = nominative, because it is the thing being talked about as the subject

You would remove the definite article from borðið:

• Borð er óhreint eftir morgunmatinn

That said, this sounds less natural in many ordinary situations, because English often uses a where Icelandic may prefer either no article or a more specific construction depending on context.

The sentence you were given, Borðið er óhreint eftir morgunmatinn, is the most natural choice for The table is dirty after breakfast.