Enginn þeirra er hér í dag.

Enginn means no one / nobody / none. In this sentence, it is the main subject.

A useful way to think of Enginn þeirra er hér í dag is:

• enginn = no one / none
• þeirra = of them
• er = is
• hér = here
• í dag = today

So the structure is very close to none of them is here today.

Also, enginn changes form for gender, number, and case. The basic nominative singular forms are:

• enginn — masculine
• engin — feminine
• ekkert — neuter

Here, enginn is used because it refers to people in a general or masculine/default way.

Because þeirra is the genitive plural form of the pronoun they.

Compare:

• þeir = they
• þeirra = of them / their

After enginn, Icelandic commonly uses the genitive to express the idea of none of them. So:

• Enginn þeirra = none of them

This is one of the big differences from English: Icelandic often uses case endings where English uses a separate word like of.

Because Icelandic expresses that meaning through case, not with a separate preposition.

In English, you say:

• none of them

In Icelandic, the of them part is built into þeirra, which is the genitive form.

So Enginn þeirra is literally something like:

• no one of-them

That is completely normal Icelandic grammar.

Because the grammatical subject is enginn, and enginn is singular.

So Icelandic uses the singular verb:

• Enginn þeirra er ... = None of them is ...

Even though the meaning involves several people, the word enginn itself is singular, so the verb agrees with it.

This is similar to formal English none of them is, although everyday English often says none of them are. Icelandic normally keeps it singular here.

Er is the present-tense singular form of vera, which means to be.

So:

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

In this sentence, er matches the singular subject enginn.

Yes. It is a very common pattern.

The structure is:

• enginn + genitive plural = none of ...

Examples:

• Enginn þeirra kom. = None of them came.
• Enginn nemendanna vissi svarið. = None of the students knew the answer.

So once you recognize enginn + genitive, you can often understand it as none of ....

Hér í dag is a natural adverb order meaning here today.

In neutral Icelandic word order, place and time expressions often come after the verb, and place often comes before time:

• er hér í dag = is here today

That said, Icelandic word order is fairly flexible, especially for emphasis. For example, you could move í dag earlier if you wanted to stress today.

But in a plain, neutral sentence, Enginn þeirra er hér í dag sounds very natural.

It specifically means none of them is here today, because of þeirra.

That tells you the speaker is talking about a particular group already known from context.

Compare:

• Enginn er hér í dag. = Nobody is here today.
• Enginn þeirra er hér í dag. = None of them is here today.

So þeirra makes the meaning more specific.

Sometimes, yes, but it depends on what kind of people you are referring to.

• enginn is masculine singular
• engin is feminine singular
• ekkert is neuter singular

If the implied person is feminine, you may see engin þeirra. But enginn is often used as the default when referring to people in a mixed or unspecified group.

So in a general sentence like this, enginn þeirra is the safest form to learn first.

Í dag is the normal Icelandic expression for today.

Literally, it is:

• í = in
• dag = day

But as a fixed everyday expression, it simply means today. You should learn it as a phrase.

Examples:

• Ég vinn í dag. = I am working today.
• Hún kemur ekki í dag. = She is not coming today.

So in your sentence, hér í dag means here today.