Smekkurinn er skítugur eftir kvöldmat.

Here, smekkurinn means the bib.

The noun smekkur can also mean taste, but that meaning does not fit this sentence. With dirty after dinner, the natural meaning is clearly bib.

Because Icelandic usually puts the definite article onto the end of the noun.

• smekkur = a bib
• smekkurinn = the bib

So -inn is the the part here. This is very common in Icelandic.

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

Here, smekkurinn is:

• masculine
• singular
• nominative

So the adjective also has to be masculine singular nominative:

• skítugur = masculine
• compare with skítug = feminine
• and skítugt = neuter

Even though the adjective comes after er, it still agrees with the subject.

Because that would be a different structure.

• smekkurinn er skítugur = the bib is dirty
• skítugi smekkurinn = the dirty bib

In your sentence, skítugur is a predicate adjective after the verb er. It is not being used directly in front of the noun.

Because eftir here takes the accusative case.

The dictionary form is:

• kvöldmatur = dinner

But after eftir in this time expression, it changes to the accusative:

• eftir kvöldmat = after dinner

This is a very common Icelandic pattern: prepositions often require a specific case.

Because Icelandic, like English, often uses meal words without the when speaking generally.

So:

• eftir kvöldmat = after dinner

This works much like English after dinner, not after the dinner, when you mean the meal in a general everyday sense.

Yes, it is one word: kvöldmatur in its base form.

It is a compound:

• kvöld = evening
• matur = food / meal

So kvöldmatur literally means evening meal.

Icelandic forms a lot of nouns this way, so compounds are very common.

The sentence has a very normal basic order:

• Smekkurinn = subject
• er = verb
• skítugur = complement
• eftir kvöldmat = time phrase

So the pattern is basically:

Subject + Verb + Adjective/Complement + Time phrase

This is also consistent with Icelandic being a verb-second language. In a simple statement like this, the verb naturally appears early in the sentence.

It is a very common everyday word for dirty.

It can feel a little more earthy than a more neutral word like óhreinn, but it is completely normal in ordinary speech, especially for things like:

• clothes
• hands
• shoes
• a bib

So in this sentence, skítugur sounds natural.