Jeg går til boghandlen i byen, hvis jeg ikke finder bogen på biblioteket.

Because Danish makes compound nouns very freely.

• bog = book
• handel = trade/shop
• boghandel = bookshop / bookstore
• boghandlen = the bookshop / bookstore

English often writes similar ideas as separate words, but Danish usually joins them into one word. This is very common:

• bibliotekskort = library card
• togstation = train station
• sommerferie = summer holiday

So boghandlen is not unusual at all; it is the normal Danish spelling.

Those endings are the definite article in Danish. Instead of putting the in front, Danish usually attaches it to the noun.

• en bog = a book

• bogen = the book

• en boghandel = a bookstore

• boghandlen = the bookstore

• et bibliotek = a library

• biblioteket = the library

• en by = a town/city
• byen = the town/city

So English uses a separate word, but Danish often uses an ending.

Because Danish often uses the present tense for future meaning when the context makes the time clear.

Here, the condition with hvis shows that this is about what will happen later:

• Jeg går til boghandlen ... hvis jeg ikke finder bogen ...

A native English speaker may expect something like I will go or if I don't find, but Danish very often just uses the present tense in this kind of sentence.

You could add vil for emphasis in some contexts, but it is not necessary.

Strictly speaking, gå usually means walk.

So many learners will naturally understand jeg går til boghandlen as I walk to the bookstore.

That said, in simple learner examples, gå is sometimes used a bit more loosely where English would say go. But if you want to be very precise, gå does suggest going on foot.

A more transport-neutral wording might use another verb, depending on context.

These are three different prepositions with different jobs.

• til boghandlen: til is used for movement to/toward a place.
• på biblioteket: på is often used with institutions and places where you are at that place in a functional sense.
• i byen: i means in the town/city.

A useful contrast is this:

• Jeg går til boghandlen = I go/walk to the bookstore.
• Jeg er i boghandlen = I am in the bookstore.
• Jeg finder bogen på biblioteket = I find the book at the library.

With bibliotek, Danish often prefers på biblioteket when talking about being at the library as a place/institution, not just physically inside the building.

Because this is a subordinate clause introduced by hvis.

In Danish, word order changes in subordinate clauses. The negative word ikke usually comes before the finite verb there:

• hvis jeg ikke finder bogen
• fordi jeg ikke har tid
• når hun ikke kommer

But in a main clause, ikke usually comes after the finite verb:

• Jeg finder ikke bogen
• Hun har ikke tid

So this is a very important pattern:

• main clause: verb + ikke
• subordinate clause: ikke
  • verb

Because hvis introduces a subordinate clause, and subordinate clauses do not follow the normal Danish main-clause V2 pattern.

In a main clause, Danish often puts the finite verb in second position:

• Jeg finder ikke bogen.
• I dag finder jeg ikke bogen.

But after hvis, the clause is subordinate, so the structure is more like:

• hvis + subject + ikke + verb + ...

That is why you get:

• hvis jeg ikke finder bogen

not a main-clause-style inversion.

Use hvis for a condition:

• Jeg går til boghandlen, hvis jeg ikke finder bogen på biblioteket.
  • This means: under that condition, I will go.

Use om when English means whether:

• Jeg ved ikke, om bogen er på biblioteket.
  • I don’t know whether the book is at the library.

So a good shortcut is:

• hvis = conditional if
• om = whether / if

Often i byen means something like in town rather than a very literal inside the city.

It can refer to:

• the local town/city
• the town center
• the place where the shops are

So in this sentence, boghandlen i byen is naturally understood as the bookstore in town or the bookstore in the city/town center, depending on context. Danish often uses byen this way when the place is already understood from the situation.