Jeg tæller bøgerne på hylden.

Because bøgerne is the definite plural form: the books.

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

Danish often puts the definite ending directly on the noun instead of using a separate word like the in English.

Also, this noun changes vowel in the plural:

• bog → bøger

So bøgerne is not just bog + erne in a simple mechanical way you can always guess; it is a real plural form you need to learn.

Because hylden is the definite singular form of en hylde.

• en hylde = a shelf
• hylden = the shelf

So på hylden means on the shelf.

Just like with bøgerne, Danish usually marks definiteness by adding an ending to the noun.

In Danish, the definite article is very often attached to the end of the noun.

Compare:

• English: the shelf

• Danish: hylden

• English: the books
• Danish: bøgerne

This is one of the biggest differences from English. Instead of putting the before the noun, Danish often adds:

• -en / -et for singular definite
• -ne for plural definite

Tæller means count or am counting here.

Its infinitive, or dictionary form, is at tælle = to count.

In this sentence:

• Jeg tæller = I count / I am counting

Danish present tense is often used where English might use either a simple present or a present continuous form, depending on context.

So Jeg tæller bøgerne på hylden can mean:

• I count the books on the shelf
• I am counting the books on the shelf

Because Danish verbs usually do not change by person the way English verbs sometimes do.

For example:

• jeg tæller = I count
• du tæller = you count
• han tæller = he counts
• vi tæller = we count

The verb form stays the same in the present tense. That is much simpler than English.

Because på means on, and a shelf is usually thought of as a surface things are placed on, not in.

So:

• på hylden = on the shelf

Using i would mean in, which usually does not fit the normal idea of books resting on a shelf.

This is similar to English:

• books on the shelf

not normally:

• books in the shelf

The word order is:

Jeg + tæller + bøgerne + på hylden

So:

• Jeg = subject
• tæller = verb
• bøgerne = object
• på hylden = prepositional phrase

This is a normal Danish main-clause pattern.

Also, Danish is a verb-second language in main clauses. That means the finite verb normally comes in the second position.

Here the subject comes first, so the verb naturally comes right after it:

• Jeg tæller ...

Most naturally, it describes the books: the books that are on the shelf.

So the sentence is usually understood as:

• I am counting the books that are on the shelf

In real life, context decides everything, but this is the most likely reading.

Jeg means I.

Unlike English, Danish does not always capitalize the first-person pronoun. It is only capitalized here because it is the first word of the sentence.

So:

• jeg = I
• at the beginning of a sentence: Jeg

Inside a sentence, it would normally stay lowercase.

It is at least not fully regular in the plural.

You might expect something simple, but the forms are:

• en bog
• bogen
• bøger
• bøgerne

The vowel changes from o to ø in the plural, so this is something learners usually just memorize.

No, not with this word order.

Jeg tæller bøgerne på hylden clearly means I count the books on the shelf.

If tæller meant counts / matters in the sense of is important, the sentence structure and meaning would be different. In this sentence, jeg is the subject, so I am doing the counting.

You would remove the definite endings and use indefinite forms.

For example:

• Jeg tæller bøger på en hylde = I am counting books on a shelf

Compare:

• bøgerne = the books
• bøger = books

and

• hylden = the shelf
• en hylde = a shelf

So the endings are doing a lot of important work in Danish.

Yes. Tæller is present tense.

Danish often uses the simple present where English might prefer either:

• I count
• I am counting

So depending on context, Jeg tæller bøgerne på hylden can naturally be understood as I am counting the books on the shelf.