A diák üzenetet ír a tanárnak, amikor hazaér.

Yes, a/az is the Hungarian definite article (like the). Hungarian uses it more broadly than English:

• A diák = the student (often a generic student in a general statement, not necessarily a specific one)
• a tanárnak = to the teacher So the sentence can be understood as a general, typical situation: The student writes a message to the teacher when they get home.

The -t marks the accusative case (the direct object).

• üzenet = message
• üzenetet = (a) message (as the thing being written)

Because the verb ír (to write) typically takes the recipient in the dative case (-nak/-nek), meaning to someone:

• ír a tanárnak = writes to the teacher
  By contrast, -hoz/-hez/-höz means physical direction (to/towards a place/person), more like going to the teacher rather than writing to the teacher.

Hungarian uses vowel harmony for many endings. The dative is -nak/-nek:

• back vowels → -nak
• front vowels → -nek tanár has the back vowel á, so it takes -nak → tanárnak.

Hungarian has two main present-tense conjugations:

• indefinite (alanyi) when the object is indefinite or not a specific known thing
• definite (tárgyas) when the object is definite/specific (roughly: the, this, that, a name, etc.)

Here the object is üzenetet = a message (not the message), so Hungarian uses the indefinite form: ír.

If it were the message, you’d get the definite conjugation, e.g.:

• A diák megírja az üzenetet. = The student writes (finishes writing) the message.

Common triggers for the definite conjugation include:

• a definite article: az üzenetet (the message)
• demonstratives: ezt/azt az üzenetet (this/that message)
• possessives: az üzenetét (his/her message)
• proper nouns / uniquely identified things

In your sentence, üzenetet has no definite article and is interpreted as indefinite.

Hungarian word order is flexible and often reflects emphasis (topic–focus structure). A neutral, natural order here is:

• A diák (topic: who we’re talking about)
• üzenetet (object, non-emphatic)
• ír (verb)
• a tanárnak (recipient)

You can rearrange it to emphasize different parts, e.g.:

• A tanárnak ír üzenetet a diák. (emphasis on to the teacher)
• Üzenetet ír a diák a tanárnak. (emphasis on a message)

amikor means when and introduces a subordinate time clause. Hungarian normally uses a comma before subordinate clauses:

• main clause: A diák üzenetet ír a tanárnak,
• time clause: amikor hazaér.

haza- is a common “verbal prefix” meaning home(wards), and it often attaches to verbs:

• ér = arrive / reach
• hazaér = arrive home (reach home)

Hungarian often writes these as one word in this kind of straightforward use.

Yes, with nuance:

• hazaér = arrive home (simple, common)
• hazaérkezik = arrive home (slightly more formal/explicit)
• hazatér = return home (focuses more on returning)

All can work depending on style and meaning.

Hungarian often uses the present tense for habitual actions and also for future-like meaning when the time is clear from context (especially with time clauses like amikor...). So this can mean:

• a general habit: When the student gets home, they write a message...
• a planned/typical future scenario, depending on context