Min mor vil ikke sælge bøgerne; hun vil hellere donere dem eller forære dem til naboerne.

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

• bog = book
• bøger = books
• bøgerne = the books

And:

• nabo = neighbor
• naboer = neighbors
• naboerne = the neighbors

So where English uses a separate word like the, Danish often adds -en, -et, or -ne to the noun.

In a normal main clause, Danish usually puts ikke after the finite verb.

Here, the finite verb is vil.

So the pattern is:

• hun vil ikke sælge ... = she will not sell ...

This is very common in Danish word order:

• Jeg kan ikke komme. = I cannot come.
• Han vil ikke spise. = He does not want to eat.

So vil ikke sælge is the expected order.

Here vil is being used like wants to or intends to, not only future will.

So:

• Min mor vil ikke sælge bøgerne means something like My mother does not want to sell the books.

In Danish, vil often covers both ideas:

• future intention
• willingness / desire

The exact meaning depends on context.

Hellere means rather.

So:

• hun vil hellere donere dem ... = she would rather donate them ...

It shows preference between options. In this sentence, the idea is that she prefers donating or giving them away instead of selling them.

A useful comparison:

• gerne = gladly / willingly
• hellere = rather
• helst = preferably / most of all

Because Danish main clauses normally follow verb-second word order, and adverbs like hellere usually come after the finite verb.

So:

• hun vil hellere donere dem

The order is:

• subject: hun
• finite verb: vil
• adverb: hellere
• infinitive: donere

This is the normal pattern in a main clause.

They are similar, but not identical.

• donere = to donate
  Often sounds more formal and is often used for giving something to a cause, organization, collection, charity, library, etc.

• forære = to give as a gift / give away
  More everyday and personal. It often suggests giving something to a person for free.

So in this sentence:

• donere dem suggests giving the books away in a more donation-like sense
• forære dem til naboerne suggests giving them directly to the neighbors

Because dem is the object of both verbs:

• donere dem
• forære dem til naboerne

Repeating it makes the structure clear and natural. Danish often repeats the object when two verbs are linked and both need it.

If you removed one dem, the sentence would sound less clear or less complete.

Dem refers back to bøgerne.

So:

• bøgerne = the books
• dem = them

Because bøgerne is plural, the matching object pronoun is dem.

Because til means to, and naboerne is the receiver.

So:

• forære dem til naboerne = give them to the neighbors

The books are the thing being given, and the neighbors are the people receiving them.

Yes. A very common everyday alternative would be give dem til naboerne.

So these are similar:

• forære dem til naboerne
• give dem til naboerne

But forære can sound a little more like give as a gift or give away for free, while give is the more general verb to give.

The semicolon links two closely related independent clauses:

• Min mor vil ikke sælge bøgerne
• hun vil hellere donere dem eller forære dem til naboerne

A semicolon is stronger than a comma but weaker than a full stop. It shows that the two parts belong closely together.

A full stop would also be possible:

• Min mor vil ikke sælge bøgerne. Hun vil hellere donere dem eller forære dem til naboerne.

In Danish, common nouns are not capitalized, including family words like:

• mor = mother / mom
• far = father / dad

So min mor is written with a lowercase m.

Danish does not capitalize nouns the way German does, and it does not usually capitalize family titles in the middle of a sentence the way English sometimes does.

Yes, usually the neighbors in the relevant context.

Because it is definite plural, it points to neighbors that are understood from the situation. In English, we might also say the neighbors even if we have not named them specifically.

If Danish wanted to be more explicitly possessive, it could say something like sine naboer in some contexts, but naboerne works naturally here.

Yes, sælge is the infinitive.

After modal verbs like vil, kan, skal, må, and bør, Danish normally uses the bare infinitive without at.

So:

• vil sælge
• vil donere
• vil forære

Compare:

• Hun vil sælge bøgerne.
• Hun prøver at sælge bøgerne.

After prøver, you use at. After vil, you do not.

Grammatically, that is very close, but in natural English we would usually say:

• My mother doesn’t want to sell the books; she would rather donate them or give them to the neighbors.

That is because Danish vil often sounds more natural in English as wants to or would rather depending on context. So the Danish structure is straightforward, but the best English wording is often a little less literal.