Jeg vil hellere forære min niece en brugt cykel end smide den ud.

It means rather ... than ....

So:

• hellere = rather
• end = than

In this sentence, Jeg vil hellere forære min niece en brugt cykel end smide den ud means the speaker prefers one action over another:

• forære min niece en brugt cykel = give my niece a used bike
• smide den ud = throw it out

A very common pattern in Danish is:

hellere + verb/action + end + verb/action

For example:

• Jeg vil hellere blive hjemme end gå ud.
  = I would rather stay home than go out.

Here vil is best understood as part of the idea would rather.

On its own, vil can mean different things depending on context, including:

• will
• want to
• be willing to

But when you combine it with hellere, the meaning becomes very close to would rather:

• Jeg vil hellere ... = I would rather ...

So in this sentence, vil is not mainly about the future. It expresses preference.

forære means give as a gift, give away, or present to someone.

It is more specific than give.
Compare:

• give = a general give
• forære = give something to someone as a present or as a generous act

So forære min niece en brugt cykel suggests:

• the bike is being given to the niece
• it is not just being handed over temporarily
• it has the feeling of a gift or donation

A more neutral version would be:

• Jeg vil hellere give min niece en brugt cykel ...

That is possible, but forære sounds more natural if the idea is give it to her instead of throwing it away.

Because Danish often puts the recipient before the thing being given.

So:

• forære min niece en brugt cykel
• literally: give my niece a used bike

This is the same basic pattern as English:

• give someone something

Here:

• min niece = the recipient
• en brugt cykel = the thing being given

You can also express the recipient with til:

• forære en brugt cykel til min niece

That also works, but the version in the sentence is very natural and common.

Because brugt is the correct form here.

In en brugt cykel, the noun is:

• singular
• indefinite
• common gender (en-word)

brugt is a past participle used like an adjective, and in this kind of phrase it stays brugt.

Compare:

• en brugt cykel = a used bike
• et brugt bord = a used table
• brugte cykler = used bikes
• den brugte cykel = the used bike

So the -e form appears in plural and definite expressions, but not here.

Because smide is part of an infinitive structure controlled by vil.

After modal verbs such as:

• vil
• kan
• skal
• må
• bør

Danish normally uses the infinitive without at.

So:

• Jeg vil forære ...
• not Jeg vil at forære ...

The same idea continues after end:

• Jeg vil hellere forære ... end smide den ud

In other words, both actions belong to the same preference structure:

• I would rather give it away
• than throw it out

Because smide ud is a verb + particle combination, and with a pronoun object, Danish normally puts the pronoun before the particle.

So:

• smide den ud = throw it out

Here:

• smide = throw
• ud = out
• den = it

This word order is very common in Danish:

• tage den af = take it off
• skrive det ned = write it down
• smide den ud = throw it out

So den goes between the verb and the particle.

den refers back to en brugt cykel.

Danish uses:

• den for common-gender nouns
• det for neuter nouns

Since cykel is an en-word:

• en cykel
• therefore: den

So smide den ud means throw the bike out.

Because the sentence is making a comparison, not simply joining two actions.

• og = and
• end = than

The speaker is not saying:

• I would rather give my niece a used bike and throw it out

The speaker is saying:

• I would rather give my niece a used bike than throw it out

So end is the correct word because it marks contrast between two choices.

These two words can be tricky for English speakers.

• niece is pronounced roughly NEE-seh
• forære is pronounced roughly fo-REH-reh

A few notes:

• Danish niece is not pronounced like English niece
• forære has an æ sound, which is more open than English ay
• in natural speech, Danish vowels are often softer and less clearly separated than English learners expect

A rough learner-friendly version of the whole middle part would be:

• hellere forære min niece en brugt cykel

approximately:

• HEL-erə fo-REH-rə min NEE-sə en BROGT SY-gəl

That is only an approximation, but it can help you get started.