Frumsýningin er í dag.

It is a compound plus the definite article:

• frum- = first, original
• sýning = showing, screening, performance
• -in = the

So frumsýningin is literally something like the first showing/performance, which is why it means the premiere.

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

So:

• frumsýning = premiere
• frumsýningin = the premiere

This is one of the first big differences English speakers notice in Icelandic.

Here, frumsýningin is:

• feminine
• singular
• nominative
• definite

It is nominative because it is the subject of the sentence: the thing that is happening today.

The base noun frumsýning is feminine, so its definite forms follow feminine patterns.

er is the present singular form of the verb vera, which means to be.

So:

• vera = to be
• er = is

Because frumsýningin is singular, Icelandic uses er, just like English uses is with the premiere.

As a phrase, í dag means today.

Word by word:

• í = in
• dag = day (in the form used in this expression)

But you should learn í dag as a fixed everyday expression meaning today, not as a word-for-word translation.

Because dagur is the basic dictionary form, but Icelandic nouns change form depending on grammar.

For dagur (day), some forms are:

• dagur = nominative
• dag = accusative
• degi = dative

In the expression í dag, Icelandic uses the form dag. So this is just the correct grammatical form for that phrase.

Yes. That is completely natural.

Both of these work:

• Frumsýningin er í dag.
• Í dag er frumsýningin.

The difference is mostly focus:

• Frumsýningin er í dag starts with the premiere
• Í dag er frumsýningin starts with today, so it highlights the time more

When you move í dag to the front, the verb still comes second: Í dag er... This is a very common Icelandic word-order pattern.

It is capitalized only because it is the first word of the sentence.

Icelandic does not capitalize common nouns the way German does. So in the middle of a sentence, it would normally be:

• frumsýningin

not capitalized unless there is some special reason.

Usually, for this meaning, frumsýningin is the more natural choice if you mean a specific premiere that speaker and listener know about.

• Frumsýningin er í dag. = The premiere is today.

Without the article:

• Frumsýning er í dag.

that sounds more indefinite, more like A premiere is today, which is less natural in many normal contexts unless you are speaking very generally or in a headline-like style.

No. Frumsýning can be used for different kinds of first public presentations, depending on context, for example:

• a film
• a play
• a TV show
• a performance

So the exact English word may vary a little by context, but the core idea is the first public showing or performance.