Læreren skriver en hovedsætning på tavlen og derefter en bisætning under den.

Because Danish usually puts the definite article at the end of the noun.

• lærer = teacher
• læreren = the teacher

So Læreren means the teacher. This is one of the biggest differences from English, where the is a separate word.

Danish verbs do not change for person the way English sometimes does.

In the present tense, skrive becomes skriver for all persons:

• jeg skriver = I write
• du skriver = you write
• han/hun skriver = he/she writes
• vi skriver = we write

So skriver is simply the present tense form, and it works with all subjects.

Because the sentence presents a main clause and a subordinate clause as items being written, so they are indefinite:

• en hovedsætning = a main clause
• en bisætning = a subordinate clause

But tavlen is definite:

• tavle = board
• tavlen = the board

That suggests the board is already known from the situation, or it is the obvious classroom board.

You have to learn the grammatical gender of each noun. In this sentence, these nouns are all common gender nouns, so they take en:

• en lærer
• en hovedsætning
• en bisætning
• en tavle

If a noun were neuter, it would take et instead.

Yes. Danish very often builds new words by combining smaller words into one compound.

• hovedsætning = hoved

  • sætning
  • hoved = head/main
  • sætning = sentence/clause

• bisætning = bi

  • sætning
  • bi- here means something like secondary/subordinate
  • sætning = sentence/clause

So both are standard grammatical terms, and Danish writes them as one word.

Because på is the normal preposition for writing on a surface.

So:

• på tavlen = on the board

That is the natural choice when someone writes on a blackboard, whiteboard, or similar surface.

Also notice the definite form again:

• tavle = board
• tavlen = the board

Derefter means after that or then.

So og derefter means something like:

• and then
• and after that

Its position is normal here. The sentence first says what the teacher writes, and then derefter signals the next step in the sequence.

Most naturally, den refers to hovedsætning.

So the idea is that the teacher writes a subordinate clause under the main clause.

That said, an English speaker may notice that den could theoretically also refer to tavlen, because both hovedsætning and tavle are en-words. In real use, context usually makes the intended meaning clear. Here, the most likely reading is under the main clause.

Because the pronoun must match the grammatical gender of the noun it refers to.

• en-words take den
• et-words take det

Since hovedsætning is an en-word, you get under den.

Because one verb can cover both parts.

The structure is basically:

• The teacher writes a main clause on the board
• and then a subordinate clause under it

Danish, like English, does not need to repeat the verb if it is the same action. Repeating it would be possible in some contexts, but here it would sound less smooth.

Yes. Danish main clauses usually follow the verb-second pattern.

For example:

• Derefter skriver læreren en bisætning under den.

Here derefter comes first, so the verb skriver must come before the subject læreren.

In your original sentence, the subject comes first, so the order stays more straightforward:

• Læreren skriver ... og derefter ...