Kennarinn segir að fyrsti stafurinn í nafninu hennar sé hástafur.

Kennari means teacher.
Kennarinn means the teacher.

Icelandic usually puts the definite article on the end of the noun, instead of using a separate word like English the. So:

• kennari = teacher
• kennarinn = the teacher

Here að means that and introduces a subordinate clause after segir.

So the structure is:

• Kennarinn segir = the teacher says
• að ... = that ...

English often drops that in sentences like this, but Icelandic commonly keeps að.

Sé is the subjunctive form of vera (to be).

In sentences of reported speech, especially in more careful or written Icelandic, you often see the subjunctive after verbs like segja. It can suggest that the speaker is reporting someone else's statement rather than directly stating it as a plain fact.

So:

• er = is
• sé = be / is (subjunctive)

In everyday speech, many speakers may also use er here, but sé is very natural and standard in this kind of sentence.

This means the first letter.

There are two important things happening:

1. Stafurinn is stafur (letter) with the definite article attached, so it means the letter.
2. Fyrsti (first) agrees with stafurinn in gender, number, and case.

Because stafur is masculine singular nominative here, the ordinal appears as fyrsti.

So:

• stafur = letter
• stafurinn = the letter
• fyrsti stafurinn = the first letter

Because the preposition í can take different cases depending on meaning:

• accusative for motion into
• dative for location in

Here the meaning is location: the letter is in the name, not moving into it. So Icelandic uses the dative.

The noun is:

• nafn = name
• nafni = name (dative singular)
• nafninu = the name (dative singular with the definite article)

Hennar means her in the possessive sense, as in her name.

In Icelandic, third-person possessive forms like hennar, hans, and þeirra usually come after the noun:

• nafnið hennar = her name

Also, hennar is not in dative here because it is not the word governed by í. The preposition í governs nafninu.
Hennar shows possession, so it stays in its possessive/genitive form.

So in í nafninu hennar:

• nafninu = dative, because of í
• hennar = genitive/possessive, because it means her

Icelandic has no indefinite article. There is no separate word corresponding to English a/an.

So a bare noun like hástafur can mean:

• a capital letter
• capital letter (depending on context)

After vera (to be), predicate nouns are very often just left bare, so:

• sé hástafur = is a capital letter

Hástafur means capital letter or uppercase letter.

It is a compound noun:

• stafur = letter
• hástafur = capital/uppercase letter

A useful comparison is:

• hástafur = uppercase letter
• lágstafur = lowercase letter

Icelandic uses compounds like this very often, so this is a good word-building pattern to notice.