Während die anderen mitschreiben, höre ich erst zu und notiere dann nur die wichtigsten Punkte.

Here, während is a subordinating conjunction meaning while.

So:

• Während die anderen mitschreiben ... = While the others are taking notes ...

This is different from während used as a preposition, where it means during and is followed by a noun phrase, for example:

• während des Unterrichts = during the lesson

In your sentence, because während introduces a full clause with a subject and verb, it is a conjunction.

Because während introduces a subordinate clause, and in German subordinate clauses, the conjugated verb usually goes to the end.

So the normal main-clause order would be something like:

• Die anderen schreiben mit.

But after während, it becomes:

• Während die anderen mitschreiben

That final position for the verb is one of the most important patterns in German word order.

Because the whole subordinate clause Während die anderen mitschreiben takes up the first position in the sentence. In a German main clause, the finite verb must come second.

So the structure is:

1. Während die anderen mitschreiben
2. höre
3. ich
4. the rest of the sentence

That is why German says:

• Während die anderen mitschreiben, höre ich ...

and not:

• Während die anderen mitschreiben, ich höre ...

This is a very common pattern after any fronted element, especially a subordinate clause.

Mitschreiben means something like:

• to write along
• to take notes as someone is speaking
• to copy down what is being said

The prefix mit- adds the idea of doing something along with someone or something else.

So schreiben is just to write, but mitschreiben suggests writing things down as you listen, often in class or a meeting.

Because the verb is zuhören, which is a separable verb.

Its base form is:

• zuhören = to listen

In a main clause, separable verbs split:

• Ich höre zu.

So in your sentence:

• höre ich erst zu

The conjugated part höre stays in the normal verb position, and the prefix zu moves to the end.

If the verb were in an infinitive or in a subordinate clause, it would stay together:

• Ich will zuhören.
• ..., weil ich zuhöre.

A useful extra point: zuhören normally takes a dative object if you say who you are listening to:

• Ich höre dem Lehrer zu.

In your sentence, that object is simply left unstated.

Here, erst ... dann means first ... then.

So the sequence is:

1. I listen first
2. then I write down only the most important points

In other contexts, erst can sometimes mean only, but here that is not the meaning. The pairing with dann makes the sequence clear.

So:

• erst zuhören = listen first
• dann notieren = then note down

Yes, you can think of it as an omitted noun.

Die anderen literally means the others, but in German it often stands on its own without saying exactly the other students, the other people, and so on.

So it can mean:

• the others
• the other people
• the other students

Grammatically, die anderen is the subject of the subordinate clause, so it is in the nominative plural.

It is the accusative plural object of notiere.

You are noting down what?

• die wichtigsten Punkte

Because Punkte is plural, the article is die in both nominative and accusative plural. The adjective wichtigsten has the weak ending -en because it follows the definite article die.

So:

• die = definite article, plural
• wichtigsten = superlative adjective, weak ending
• Punkte = plural noun

The phrase means the most important points.

Because German uses a comma to separate a subordinate clause from a main clause.

So:

• Während die anderen mitschreiben, höre ich ...

That comma is required.

German punctuation is stricter than English here: when a conjunction like während, weil, dass, or obwohl introduces a subordinate clause, the clause is normally set off with a comma.

German uses the present tense much more broadly than English does. The simple present can cover both:

• habitual meaning: I usually do this
• current ongoing meaning: I am doing this right now

So:

• die anderen mitschreiben
• höre ich zu
• notiere

can all describe an action happening now, without needing a special progressive form like English am listening or are taking notes.

German does have ways to emphasize ongoing action, but in normal speech the present tense is usually enough.