Learn Hardcore Danish: Hans mor laver te, mens hun venter på et vigtigt telefonopkald. - His mother makes tea while she waits for an important phone call.

Hans as a name:
Then Hans mor = Hans’s mother (the mother of a man called Hans).
hans as a possessive pronoun (his):
Then hans mor = his mother (someone’s mother, not necessarily named Hans).

The only difference in writing is the capital letter:

Hans (capital H) = the name Hans.
hans (small h) = the pronoun his.

In your sentence it’s written Hans, so the most natural reading is Hans’s mother.

Why is there no article in Hans mor? Why not Hansen mor or Hans’s mor?

Danish shows possession differently from English:

You don’t add ’s after a proper name in front of a common family member word.
- English: Hans’s mother
- Danish: Hans mor (not Hans’s mor, not Hansens mor)
You also don’t use an article in this pattern:
- English: the mother of Hans → Hans’s mother
- Danish: Hans mor (no en/et)

So:

Hans mor = Hans’s mother / his mother
hans mor = his mother (some male person’s mother)

What does laver mean here, and why not gør?

Both laver and gør can mean do/make, but they’re used differently:

laver = makes, prepares, creates
- laver te = makes tea, prepares tea
- laver mad = cooks (food)
gør is more like do, perform, cause, often with adjectives or abstract things:
- gør rent = do the cleaning
- gør ondt = hurts (literally: does pain)

With tea, Danish normally uses laver:

Hun laver te = She is making tea.

Gør te would sound wrong or at least very unusual.

Danish uses laver and venter in the present tense, but English says is making and is waiting. Why?

Danish normally uses the simple present to cover both:

habitual/regular actions and
actions happening right now.

So:

Hun laver te
= She makes tea (in general)
= She is making tea (right now) — context decides.
Hun venter
= She waits / She is waiting.

English has a special progressive form (is making, is waiting); Danish doesn’t need that form. Danish can build a progressive-like phrase (er ved at lave te = is in the process of making tea), but in many everyday sentences the plain present is enough and is translated into English as is doing.

Why is there a comma before mens in Hans mor laver te, mens hun venter…?

Because mens hun venter på et vigtigt telefonopkald is a subordinate clause (a dependent clause):

Main clause: Hans mor laver te
Subordinate clause (introduced by mens): mens hun venter på et vigtigt telefonopkald

In modern Danish comma rules (which most learners are taught):

You put a comma before most conjunctions that introduce a subordinate clause, including mens, fordi, når, da, selvom, etc.

So the comma marks the boundary between the main clause and the subordinate clause:

Hans mor laver te, mens hun venter på et vigtigt telefonopkald.

What exactly does mens mean, and how does it affect the word order in mens hun venter på…?

mens means while (at the same time as):

mens hun venter = while she waits / is waiting.

In Danish, mens introduces a subordinate clause, and subordinate clauses have a slightly different typical word order from main clauses:

Main clause (V2 word order):
- Hun venter på et vigtigt telefonopkald.
  (subject hun, then verb venter)
Subordinate clause after mens:
- mens hun venter på et vigtigt telefonopkald
  (conjunction mens, then subject hun, then verb venter)

So the basic pattern here is:

mens + subject + verb + rest of the clause
Which matches mens hun venter på et vigtigt telefonopkald.

Why is it venter på and not just venter or venter for?

The verb at vente behaves differently in Danish than to wait in English:

vente på noget / nogen = wait for something / someone
- Hun venter på et vigtigt telefonopkald.
  = She is waiting for an important phone call.
vente without a preposition usually means simply wait, with no direct object:
- Hun venter. = She is waiting.

You don’t say venter for in Danish. When you mean wait for, you almost always say venter på:

vente på bussen = wait for the bus
vente på dig = wait for you
vente på svar = wait for an answer

Why is it et vigtigt telefonopkald and not en? How do I know which article to use?

Danish has two grammatical genders:

common gender (en-words)
neuter gender (et-words)

The noun (et) opkald (a call) is neuter, so it takes:

the indefinite article et
the adjective in neuter singular form vigtigt (with -t)

That gives:

et vigtigt opkald = an important call
and with telefon- in front: et vigtigt telefonopkald.

If the noun were common gender, you would use en and usually vigtig:

en vigtig bog (a(n) important book)
et vigtigt brev (an important letter)

So the gender of opkald (neuter) controls both et and the -t on vigtigt.

What kind of word is telefonopkald, and can it be split up?

telefonopkald is a compound noun:

telefon = telephone, phone
opkald = a call (literally a calling up)

Danish often joins nouns together into one long word. In writing, you do not separate the parts:

telefonopkald (correct)
telefon opkald (incorrect as standard spelling)

Grammatically, telefonopkald behaves like opkald:

It is neuter: et telefonopkald, det telefonopkald
Plural: telefonopkald (same form), de telefonopkald.

Why is it hun and not hende in mens hun venter?

In Danish:

hun = she (subject form)
hende = her (object / after prepositions)

You use hun as the subject of a verb:

Hun venter. = She is waiting.
Hun laver te. = She makes tea.

You use hende as an object:

Jeg ser hende. = I see her.
Jeg venter på hende. = I am waiting for her.

In mens hun venter på et vigtigt telefonopkald, hun is the subject of venter, so hun (not hende) is correct.

Who does hun refer to in this sentence? The mother or someone else?

By default, hun refers back to the nearest suitable feminine noun phrase in the context, which here is Hans mor:

Hans mor laver te, mens hun venter på et vigtigt telefonopkald.

So the natural reading is:

Hans’s mother is making tea while she (Hans’s mother) waits for an important phone call.

If it were meant to refer to some other she, that would normally have to be clear from the broader context, or the sentence would be rephrased to avoid ambiguity.