Dóttir mín lítur mjög glöð út í dag.

In Icelandic, possessives are often placed after the noun in a neutral, natural way:

• dóttir mín = my daughter
• vinur minn = my friend
• móðir mín = my mother

You can also put the possessive first, as in mín dóttir, but that often sounds more emphatic, contrastive, or stylistically marked.

So here dóttir mín is the normal, straightforward way to say my daughter.

It is in the nominative singular.

Why? Because dóttir mín is the subject of the sentence, and subjects are normally in the nominative.

Also:

• dóttir is a feminine singular noun
• mín agrees with it in gender, number, and case

So mín is in its feminine nominative singular form here.

Líta is the infinitive, meaning to look.

In the sentence, the verb has to be conjugated to match the subject:

• ég lít = I look
• þú lítur = you look
• hún lítur = she looks

Since dóttir mín is a third-person singular subject, Icelandic uses lítur.

So:

• líta = to look
• lítur = looks

Because líta út is a multi-word verb meaning to look / to appear.

The basic dictionary form is:

• líta út

But in a normal Icelandic main clause, the finite verb comes early in the sentence, while the particle út often appears later:

• Dóttir mín lítur mjög glöð út í dag.

So this is not two unrelated words. It is the verb líta út, just split by the sentence structure.

A useful way to think of it is like a separable particle verb.

It comes after the verb because it is a predicate adjective, not an adjective directly attached to the noun.

Compare:

• glöð dóttir = a happy daughter
  
  • adjective directly before the noun
• dóttir mín lítur glöð út = my daughter looks happy
  
  • adjective describes the subject through the verb

Here glöð describes dóttir mín, so it must agree with that noun:

• feminine singular nominative → glöð

Other forms are:

• masculine: glaður
• feminine: glöð
• neuter: glatt

So glöð is used because dóttir is feminine singular nominative.

Mjög means very.

It modifies the adjective glöð:

• glöð = happy
• mjög glöð = very happy

So:

• lítur mjög glöð út = looks very happy

Also, mjög does not change form. It is an adverb, so it stays the same.

Í dag means today.

Literally, it is something like in day, but as a whole it is just the normal Icelandic expression for today.

So it is best learned as a fixed phrase:

• í dag = today

Even though English uses one word, Icelandic uses this prepositional expression.

Icelandic usually expresses the with a suffix attached to the noun, not with a separate word like English.

However, with possessive expressions, Icelandic often does not need an extra article in the same way English does. So dóttir mín already naturally means my daughter.

This is especially common with family terms.

You may sometimes also see forms with the suffixed article in other contexts, but here dóttir mín is perfectly normal.

Yes. Icelandic word order is fairly flexible, but it follows an important verb-second pattern in main clauses.

For example, you can move í dag to the front:

• Í dag lítur dóttir mín mjög glöð út.

That still means the same thing: My daughter looks very happy today.

Notice what happens:

• first element: Í dag
• second element: lítur

So the finite verb lítur stays in the second position.

That is why Icelandic learners often have to watch word order carefully.