Jeg smiler, hver gang jeg ser månen gennem vinduet.

Hver gang means every time or whenever. It introduces a repeated situation.

So Jeg smiler, hver gang jeg ser månen gennem vinduet means that this is something the speaker does repeatedly, not just once.

Danish uses the present tense for habits and repeated actions, just like English.

• smiler = smile / am smiling
• ser = see / am seeing

In this sentence, the meaning is habitual: every time the speaker sees the moon through the window, the speaker smiles.

Also, the Danish present tense is usually formed with -r:

• at smile → smiler
• at se → ser

Because hver gang introduces a subordinate clause. In Danish subordinate clauses, the usual word order is subject + verb:

• jeg ser
• not ser jeg

So after hver gang, jeg ser månen gennem vinduet is the normal structure.

Then the main clause changes word order because Danish main clauses follow the verb-second rule.

You would say:

Hver gang jeg ser månen gennem vinduet, smiler jeg.

Notice the main clause is now smiler jeg, not jeg smiler. That happens because the subordinate clause has taken the first position in the sentence.

In Danish, the definite article is usually attached to the end of the noun instead of being a separate word.

So:

• måne = moon

• månen = the moon

• vindue = window
• vinduet = the window

This is a very common feature of Danish nouns.

Because Danish nouns have grammatical gender.

• måne is a common gender noun, so the definite ending is -en → månen
• vindue is a neuter noun, so the definite ending is -et → vinduet

A learner usually has to memorize whether a noun is common gender or neuter.

Gennem means through.

In this sentence, gennem vinduet means through the window.

You may sometimes hear igennem, which can also mean through, but gennem is the most neutral and natural choice here. Igennem can sound a bit more emphatic or more focused on passing all the way through something, depending on context.

So for this sentence, gennem is the best default choice.

It separates the main clause from the subordinate clause:

• main clause: Jeg smiler
• subordinate clause: hver gang jeg ser månen gennem vinduet

In Danish, comma use before subordinate clauses depends partly on which comma system is being followed. Many writers include the comma here, and many learners will often see it written this way. You may also see the sentence without the comma in other correct Danish writing.

Yes, you could say:

Jeg smiler, når jeg ser månen gennem vinduet.

That would mean I smile when/whenever I see the moon through the window.

The difference is that hver gang is a little more explicit and emphatic: it clearly means every single time. Når can also express repeated meaning, but it is slightly less specific.

Gennem vinduet means through the window, which usually sounds like a specific, understood window in the situation.

Using the definite form is natural when the speaker has a particular window in mind, or when the window is obvious from context.

• gennem vinduet = through the window
• gennem et vindue = through a window

You cannot normally say gennem vindue here, because Danish countable singular nouns usually need some kind of determiner or definite marking.