Min datter løber ud i skolegården, når frikvarteret begynder.

Because datter is a common-gender noun in Danish.

Danish possessives change according to the gender and number of the noun:

• min = my, for common gender
• mit = my, for neuter
• mine = my, for plural

So you get:

• min datter = my daughter
• mit barn = my child
• mine børn = my children

Løber is the present tense of løbe = to run.

In Danish, present tense is often made by adding -r to the infinitive:

• løbe → løber
• begynde → begynder

So Min datter løber ud i skolegården means My daughter runs / is running out into the schoolyard, depending on context.

Danish present tense can often cover both simple present and present progressive English meanings.

Because the sentence expresses both:

• direction outward = ud
• movement into a place = i

So løber ud i skolegården means something like:

• runs out into the schoolyard

If you only said løber i skolegården, it would more likely mean runs in the schoolyard. If you only said løber ud, it would mean runs out, but not say where to.

So ud i is a very natural combination when someone moves outward and into a space.

Because Danish very often forms compound nouns as a single word.

Here:

• skole = school
• gård = yard/courtyard
• skolegård = schoolyard

Then the definite ending is added:

• skolegården = the schoolyard

This is very normal in Danish. English often writes these as two words or with separate structure, but Danish usually prefers one combined noun.

The ending -en is the definite article for many common-gender nouns. It corresponds to English the.

So:

• en skolegård = a schoolyard
• skolegården = the schoolyard

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

For the same reason: -et is the definite ending for many neuter nouns.

So:

• et frikvarter = a recess / a break
• frikvarteret = the recess / the break

Here Danish is talking about a specific recess period, so the definite form is natural.

Når is used for when / whenever in situations that are habitual, repeated, or generally true.

That fits this sentence well, because it describes something that happens regularly:

• Min datter løber ud i skolegården, når frikvarteret begynder.
• My daughter runs out into the schoolyard when recess begins.

Compare:

• når = when / whenever, usually repeated or expected
• da = when, for a specific event in the past
• hvis = if

So this sentence uses når because it describes a regular school routine, not a one-time past event and not a condition.

It can suggest either, and in many cases Danish når covers both ideas.

In this sentence, it most naturally means something like when / whenever recess starts, because the action is probably habitual.

So the sentence can be understood as:

• When recess begins, my daughter runs out into the schoolyard
• or Whenever recess begins, my daughter runs out into the schoolyard

English may choose one or the other depending on style, but Danish is comfortable with når here.

The dictionary form is begynde, which means to begin or to start.

So:

• begynde = infinitive
• begynder = present tense

Examples:

• Frikvarteret begynder nu = Recess begins now
• Det begynder snart = It starts soon

Yes. If the sentence begins with the subordinate clause, the main clause usually has verb-second word order, so the verb comes before the subject.

Original order:

• Min datter løber ud i skolegården, når frikvarteret begynder.

With the når-clause first:

• Når frikvarteret begynder, løber min datter ud i skolegården.

Notice the change:

• not min datter løber
• but løber min datter

This is a very important Danish word order pattern.

A few parts may be difficult for English speakers:

• min: the i is short, roughly like min but tenser
• datter: the a is quite open; the tt is not strongly aspirated like English t
• løber: the ø is a rounded vowel that English does not really have
• skolegården: the å sounds roughly like the vowel in British more for many learners, though not exactly
• begynder: the gy combination can be tricky; the y is another rounded front vowel

Very roughly, for learner purposes:

• løber ≈ LUR-ber, but with a rounded vowel instead of English u
• skolegården ≈ SKO-le-gor-den, but with Danish vowel qualities
• begynder ≈ be-GYN-er, with a Danish y

The exact sounds are best learned by listening, especially for ø and y, since English does not have direct equivalents.