Det her er den bog, som jeg vil læse i aften.

Det her means this.

A few useful points:

• det her is the very common, everyday way to say this
• dette means the same thing, but sounds more formal, written, or careful
• her literally means here, but in this pattern it helps form the demonstrative:
  • det her = this
  • det der = that

So this sentence uses a very natural spoken-style Danish form.

Because det here is not an article matching bog. It is the normal pronoun used in a sentence like This is ... / It is ....

Danish often uses det er when identifying something:

• Det er min bog. = It is my book.
• Det her er den bog ... = This is the book ...

So even though bog is common gender and normally goes with den, the sentence still starts with det her er.

If the demonstrative stood directly before the noun, then you would match the gender:

• den her bog = this book

Because Danish does not use both forms together here.

You can have:

• bogen = the book
• den bog = a definite noun phrase in this kind of structure

But not:

• den bogen ✗

So when den comes before the noun, the noun stays in its basic form bog.

Because som introduces a relative clause, while at introduces a different kind of clause.

Here, som refers back to den bog:

• den bog, som jeg vil læse i aften
• literally: the book that I will read tonight

Compare:

• som = that / who / which in a relative clause
• at = that in a content clause, like I know that...

Example with at:

• Jeg ved, at han kommer. = I know that he is coming.

So English that can correspond to different Danish words. In this sentence, it has to be som, not at.

Because in this relative clause, the missing element is the object, not the subject.

In jeg vil læse [bogen] i aften:

• jeg = subject
• bogen = object

When the relative word stands for the object, som is the normal choice:

• den bog, som jeg vil læse i aften

der is typically used when the relative word is the subject:

• den bog, der ligger på bordet = the book that is lying on the table

So:

• som: the book that I read
• der: the book that lies on the table

Because this is a subordinate clause (a relative clause), and Danish word order changes in subordinate clauses.

In a main clause, Danish usually has verb-second word order:

• I aften vil jeg læse bogen.

But after som, the clause is subordinate, so the normal order is:

• som jeg vil læse i aften

That means:

• subject first: jeg
• finite verb after that: vil
• infinitive: læse

So som vil jeg læse would sound wrong here.

Vil læse means something like will read, am going to read, or intend to read, depending on context.

The verb ville can express:

• future
• intention
• willingness

In this sentence, the natural meaning is a future plan or intention:

• jeg vil læse i aften = I will / am going to read tonight

So it is not only a mechanical future marker. It often has a slight sense of intention as well.

I aften means tonight or this evening.

It is a fixed time expression, so it is best learned as a whole phrase.

Useful comparisons:

• i aften = tonight / this evening, one specific evening
• om aftenen = in the evening, usually in a general or habitual sense

For example:

• Jeg læser den i aften. = I’m reading it tonight.
• Jeg læser ofte om aftenen. = I often read in the evening.

So i here is just part of the normal expression i aften.

Often, yes.

In everyday speech and in less formal writing, Danish can drop som when it is the object, as it is here:

• den bog, jeg vil læse i aften

That is still very natural.

Keeping som is also completely correct and is very common in standard written Danish:

• den bog, som jeg vil læse i aften

So both are possible, but som makes the structure especially clear.

It depends on which Danish comma system is being used.

You may see both:

• den bog, som jeg vil læse i aften
• den bog som jeg vil læse i aften

The version with the comma is very common and widely taught. The version without it is also accepted in modern Danish under the new comma system.

So for a learner:

• the comma before som is very common and safe
• just try to be consistent in whatever comma style you use