Í dag lærðum við ný samheiti og andheiti, og kennarinn skrifaði þau í gæsalöppum á töfluna.

Because Icelandic main clauses usually follow the V2 rule: the finite verb goes in the second position.

Here, Í dag is placed first, so the verb lærðum has to come next, and the subject við comes after it.

Compare:

• Við lærðum ný samheiti og andheiti í dag.
• Í dag lærðum við ný samheiti og andheiti.

Both are correct, but the second one gives extra emphasis to Í dag.

Lærðum is the past tense, 1st person plural form of læra, which means to learn.

So:

• læra = to learn
• við lærðum = we learned

It matches the subject við.

They are in the accusative, because læra normally takes a direct object in the accusative: you learn something.

So in this sentence, the things learned are:

• ný samheiti
• andheiti

Both are objects of lærðum.

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

Both samheiti and andheiti are neuter plural here, and the correct strong form of nýr for neuter plural nominative/accusative is ný.

So:

• masculine plural: different form
• feminine plural: different form
• neuter plural: ný

That is why ný samheiti og andheiti is correct.

These nouns can look a little confusing because their form is often the same in singular and plural, especially in the nominative and accusative.

In this sentence, we know they are plural because:

• the meaning clearly refers to more than one
• the adjective ný fits plural as well
• later the pronoun þau refers back to them as a plural group

So even though the noun form itself does not show much change, the sentence structure tells you they are plural.

Kennarinn means the teacher.

It is made from:

• kennari = teacher
• -inn = the definite article the

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

So:

• kennari = a teacher / teacher
• kennarinn = the teacher

Because og is joining two full clauses:

• Í dag lærðum við ný samheiti og andheiti
• kennarinn skrifaði þau í gæsalöppum á töfluna

Each part has its own verb, and the second part has its own subject, kennarinn. In Icelandic, it is very normal to use a comma before og in this kind of sentence.

Because þau is the neuter plural pronoun, and it refers back to samheiti og andheiti, which are neuter nouns.

Icelandic pronouns follow grammatical gender, not natural gender. Since the things being referred to are grammatically neuter plural, the pronoun is þau.

Here it means them.

Í gæsalöppum means in quotation marks.

A few useful points:

• gæsalappir is the Icelandic word for quotation marks
• literally, it means something like goose feet
• after í, the dative is used when the meaning is location or state rather than motion

So here:

• í gæsalöppum = in quotation marks

The form gæsalöppum is the dative plural form.

Because Icelandic often uses:

• á + accusative for movement onto something
• á + dative for location on something

Here, the teacher wrote the words onto the board, so Icelandic uses á töfluna.

Compare:

• skrifaði á töfluna = wrote onto the board
• er á töflunni = is on the board

So the accusative shows direction or result of movement.

Yes. That is also correct.

The difference is mainly emphasis:

• Í dag lærðum við ... puts focus on today
• Við lærðum ... í dag is more neutral and straightforward

Both are natural Icelandic.

Because in normal Icelandic word order, a direct object pronoun often comes fairly early in the clause, before many adverbial or prepositional phrases.

So this order is very natural:

• kennarinn skrifaði þau í gæsalöppum á töfluna

The object þau comes first, and then the sentence adds more information about how and where the teacher wrote them.