Die Besprechung dauert eine halbe Stunde.

Besprechung is a feminine noun in German, so its basic article is die.

In this sentence, die Besprechung is also the subject of the sentence:

• Die Besprechung = the meeting
• dauert = lasts
• eine halbe Stunde = half an hour

So die here is both:

• the normal article for a feminine noun, and
• the nominative form used for the subject.

The infinitive is dauern = to last.

But in a normal sentence, German verbs are conjugated:

• ich dauere
• du dauerst
• er/sie/es dauert
• wir dauern
• ihr dauert
• sie dauern

Since die Besprechung is third person singular (it / she grammatically), the verb becomes dauert.

So:

• Die Besprechung dauert ... = The meeting lasts ...

In German, durations are often expressed without a preposition.

English:

• The meeting lasts for half an hour.

German:

• Die Besprechung dauert eine halbe Stunde.

So German does not need a separate word like for here. The duration is simply given directly after the verb.

This is very common with time expressions of duration:

• Der Film dauert zwei Stunden.
• Der Unterricht dauert 45 Minuten.

It is in the accusative.

With verbs like dauern, the length of time is commonly expressed in the accusative of duration.

So:

• eine halbe Stunde = accusative time expression

In this particular phrase, the form looks the same as the nominative because Stunde is feminine:

• nominative: eine halbe Stunde
• accusative: eine halbe Stunde

So the case is accusative, even though nothing visibly changes.

Because Stunde is a feminine noun.

The article must match the noun:

• masculine: ein
• feminine: eine
• neuter: ein

Since it is die Stunde, you need:

• eine Stunde
• eine halbe Stunde

Also, halbe is an adjective here, and it takes the correct ending after eine.

Because half an hour refers to one half of a single hour, so German uses the singular:

• eine halbe Stunde

Compare:

• eine Stunde = one hour
• eine halbe Stunde = half an hour

But with larger amounts, German may use the plural:

• zwei Stunden = two hours
• anderthalb Stunden = one and a half hours

So:

• eine halbe Stunde → singular
• anderthalb Stunden → plural

Halbe comes from halb = half.

Here it is being used like an adjective before a noun:

• eine halbe Stunde

Because it follows eine, it needs an adjective ending:

• eine halbe Stunde
• eine halbe Minute

So halb changes to halbe to fit the grammar of the phrase.

Often yes in meaning, but there are some differences in usage.

Besprechung is a normal German word meaning something like:

• meeting
• discussion meeting
• briefing

You may also hear:

• das Meeting = a borrowed word from English, very common in business contexts
• die Sitzung = session, formal meeting
• der Termin = appointment, scheduled time, not exactly the same thing

So Die Besprechung dauert eine halbe Stunde is a very natural standard-German sentence.

Yes, absolutely.

Both are natural:

• Die Besprechung dauert eine halbe Stunde.
• Die Besprechung dauert 30 Minuten.

The first sounds a bit more like everyday phrasing with half an hour. The second is more exact and direct with 30 minutes.

Yes. In a normal German main clause, the conjugated verb usually comes in the second position.

So the standard order is:

• Die Besprechung / dauert / eine halbe Stunde

This is the most neutral version.

You can change the order for emphasis, but then the verb still stays in second position:

• Eine halbe Stunde dauert die Besprechung.

That version is grammatical, but less neutral and more marked. For a learner, the basic sentence order is the best one to use.

Because all nouns are capitalized in German.

So:

• die Besprechung
• die Stunde

This is a basic rule of German spelling and one of the big differences from English.