Jeg vil gøre opgaven færdig i aften.

In Danish, vil can mean both will and want to, depending on context.

In Jeg vil gøre opgaven færdig i aften, it often works like English will:
I will finish the assignment tonight.

But Danish vil can also carry a sense of intention or willingness, a bit like want to. So the sentence can suggest:
I intend to get the assignment finished tonight.

If you want a stronger idea of obligation, Danish would more likely use skal instead of vil.

Danish often uses gøre ... færdig to mean finish or complete something.

• gøre = do / make
• færdig = finished / ready

So gøre opgaven færdig literally means something like make the assignment finished, but in natural English that becomes finish the assignment.

This is a very common Danish structure:

• gøre arbejdet færdigt = finish the work
• gøre maden færdig = finish the food / finish cooking

Because færdig describes the result of what happens to opgaven.

In gøre opgaven færdig:

• opgaven is the thing being acted on
• færdig describes the state it ends up in

So the structure is: gøre + object + result word

That is why Danish says:

• gøre huset rent = clean the house
• gøre døren lukket is not natural, but the same basic idea exists in other expressions
• gøre arbejdet færdigt = finish the work

English sometimes does something similar:

• paint the door red
• wipe the table clean

opgaven is the definite form of opgave.

• en opgave = an assignment / a task
• opgaven = the assignment / the task

Danish usually adds the definite article to the end of the noun instead of putting a separate word before it.

So:

• en bog = a book
• bogen = the book
• en opgave = an assignment
• opgaven = the assignment

In this sentence, opgaven means the assignment or the task, depending on context.

i aften is the standard Danish expression for tonight / this evening.

Here:

• i does not literally mean in in the English sense
• the whole phrase i aften should be learned as a set expression

Compare:

• i dag = today
• i morgen = tomorrow
• i aftes = last night / yesterday evening
• i aften = tonight

So even though the words may look simple, it is best to memorize i aften as the normal way to say tonight.

This is normal Danish main-clause word order:

• Jeg = subject
• vil = finite verb
• gøre = infinitive
• opgaven = object
• færdig = result adjective
• i aften = time expression

So the sentence follows the common pattern:

Subject + finite verb + other elements

Danish is a V2 language, which means the finite verb usually comes in second position in main clauses.

That also means if you move i aften to the front, the verb still stays second:

• I aften vil jeg gøre opgaven færdig.

Not:

• I aften jeg vil gøre opgaven færdig.

Yes, but it means something slightly different.

• Jeg vil gøre opgaven færdig i aften = I will / I intend to finish the assignment tonight
• Jeg skal gøre opgaven færdig i aften = I have to / am supposed to finish the assignment tonight

So:

• vil = intention, willingness, future
• skal = obligation, plan, requirement

English learners often confuse these because English will and shall / have to do not line up exactly with Danish vil and skal.

You usually put ikke after the finite verb and before the infinitive part:

• Jeg vil ikke gøre opgaven færdig i aften.

That means: I will not finish the assignment tonight
or
I don’t intend to finish the assignment tonight

This placement is very typical in Danish:

• Jeg kan ikke komme. = I cannot come.
• Jeg vil ikke vente. = I will not wait.

Yes. færdig is an adjective, so it can change form in some contexts.

Common forms:

• færdig
• færdigt
• færdige

In this sentence, many speakers use færdig with opgaven, but you may also hear færdigt in similar expressions, especially when the word functions more adverbially or in fixed combinations.

For a learner, the most useful thing is to remember the whole expression:

• gøre noget færdig / færdigt = finish something

You will meet some variation in actual usage, and that is normal.

These words can be tricky for English speakers.

• gøre has a vowel sound that does not exist naturally in English for many speakers. The ø is a front rounded vowel.
• færdig contains æ, which sounds a bit like the vowel in cat, though not exactly, and the r affects the sound.

A rough learner-friendly guide:

• gøre ≈ something like GUH-rə, but with rounded lips on the first vowel
• færdig ≈ something like FAIR-dee, though that is only approximate

The exact pronunciation varies by accent, and Danish pronunciation is often less transparent than spelling. It is best to listen to native audio and repeat whole chunks:

• vil gøre
• gøre opgaven færdig
• i aften

Yes. opgave is a broad word.

Depending on context, it can mean:

• assignment
• task
• exercise
• piece of work

For example:

• in school, opgave often means assignment or exercise
• at work, it may mean task

So in your sentence, the best English translation depends on context:

• I will finish the assignment tonight
• I will finish the task tonight