Jeg vil besøge min ven i eftermiddag.

Because Danish modal verbs like vil are followed directly by the infinitive, without any word meaning to.

• Jeg vil besøge = I want to visit / I will visit
• Not: Jeg vil at besøge

This is similar to English after modal verbs:

• I will go
• I can help
• I must leave

You do not say I will to go, and Danish works the same way here.

Vil can mean either:

• want to
• will

So Jeg vil besøge min ven i eftermiddag can often be understood as:

• I want to visit my friend this afternoon
• or I will visit my friend this afternoon

Which meaning is intended depends on context.

This is one reason Danish learners sometimes compare vil with skal:

• vil often expresses desire, intention, or willingness
• skal often expresses obligation, arrangement, or something more fixed

Because vil is the finite verb, and besøge is the infinitive that goes with it.

In Danish main clauses, the finite verb usually comes in second position. Here that verb is vil:

• Jeg = subject
• vil = finite verb
• besøge = infinitive
• min ven = object
• i eftermiddag = time expression

So the structure is:

Jeg | vil | besøge min ven i eftermiddag

This is very normal Danish word order.

Min means my and is used with a singular common gender noun.

• en ven = a friend
• min ven = my friend

Danish possessives change depending on gender and number:

• min for singular common gender nouns
• mit for singular neuter nouns
• mine for plural nouns

Examples:

• min ven = my friend
• mit hus = my house
• mine venner = my friends

So ven is a common gender noun, which is why min is correct.

Because besøge is a direct transitive verb in Danish, just like visit in English.

You say:

• besøge en ven = visit a friend

You do not need a preposition before the person.

Compare:

• Jeg besøger min ven = I visit my friend

Not:

• Jeg besøger til min ven

So in this case Danish is actually very similar to English.

I eftermiddag means this afternoon.

Literally, eftermiddag is afternoon, and Danish often uses i with parts of the day in time expressions.

So:

• i eftermiddag = this afternoon
• i morgen = tomorrow / in the morning, depending on the phrase and spacing
• i aften = this evening

Even though i often means in, you should learn i eftermiddag as a fixed time expression.

Yes. Eftermiddag is one word in Danish.

This is very common in Danish, where compounds are usually written as a single word.

Examples:

• eftermiddag = afternoon
• sommerferie = summer holiday
• arbejdsdag = workday

English often separates words where Danish combines them.

Yes. That would also be a natural sentence.

The difference is roughly:

• Jeg vil besøge min ven i eftermiddag = I want to / will visit my friend this afternoon
• Jeg besøger min ven i eftermiddag = I am visiting / I visit my friend this afternoon

Without vil, the sentence sounds more like a simple statement of plan or fact. With vil, it can sound more like intention, willingness, or future meaning.

In a yes/no question, the finite verb comes before the subject:

• Vil jeg besøge min ven i eftermiddag?

But this exact sentence is not very natural unless you are asking something like Will I visit my friend this afternoon?

A more natural question would often be:

• Vil du besøge din ven i eftermiddag? = Do you want to / Will you visit your friend this afternoon?

So the pattern is:

• statement: Jeg vil besøge...
• question: Vil jeg besøge...?

The ø is important because it is a separate vowel in Danish, not just an accented o.

Besøge is roughly pronounced like beh-SUR-ye or beh-SUH-ye, depending on accent and how simplified the explanation is for English speakers, but that is only approximate.

A few useful points:

• ø does not sound like normal English o
• the g in -ge is often softened in pronunciation
• Danish pronunciation is often less phonetic than spelling suggests

So it is best to listen to native audio and repeat the whole word as a chunk: besøge.

Yes. Ven can mean friend in general, not only a male friend.

If you want to be specific:

• ven can often be used generally
• veninde specifically means a female friend

So:

• min ven = my friend
• min veninde = my female friend

In many contexts, ven is enough unless the speaker wants to specify gender.