A pincér mosolyog, amikor vizet kérek.

Because Hungarian marks the direct object with the accusative -t.

• víz = water (dictionary form)
• vizet = water (as the thing you’re asking for / requesting)
  So in vizet kérek (I ask for water), water is the object, so it gets -t.

This is a regular stem change for some nouns when adding certain suffixes.

• Base form: víz (long í)
• Accusative: vizet (short i)
  It’s something you mostly learn word-by-word (similar to irregular plurals in English). The pronunciation is still clear: vi-zet.

Because kérek is 1st person singular: I ask / I request.

• kér = he/she asks
• kérek = I ask
  In the sentence, the speaker is the one asking for water, so Hungarian uses the matching verb ending -ek.

Both exist, but they mean different things grammatically:

• kérek = indefinite conjugation (object is not a specific identified thing)
• kérem = definite conjugation (object is specific/definite)

Here, vizet is typically treated as non-specific (some water), so kérek is natural. You’d expect kérem with something specific like:

• A vizet kérem. = I’ll have the water (the one we’re talking about).
• Ezt kérem. = I’d like this.

amikor introduces a time clause meaning when and often feels like at the time when.

• amikor is common when it clearly links to a situation/time frame.
• mikor can also mean when, but it’s more common in direct questions: Mikor jössz? (When are you coming?)

In this sentence, amikor is a normal choice for when in a statement:
A pincér mosolyog, amikor vizet kérek. = The waiter smiles when(ever) I ask for water.

It can be either, depending on context, because Hungarian present tense often covers both:

• Habitual/general: He smiles whenever I ask for water.
• “Story present” / vivid narration: He smiles when I ask for water (in this situation).

If you want to force a one-time past meaning, you’d typically use past tense:

• A pincér mosolygott, amikor vizet kértem. = The waiter smiled when I asked for water.

Hungarian often uses present tense for general truths, habits, and also for describing events in a lively way. So present-present is normal here.

If you want different time meanings, you change the tense accordingly:

• Past: mosolygott / kértem
• Future-ish intention often still uses present: holnap vizet kérek (tomorrow I’ll ask for water)

• mosolyog = (he/she) smiles (finite verb, used in sentences)
• mosolyogni = to smile (infinitive)

Hungarian sentences need a conjugated verb as the main verb, so you use mosolyog here.

Yes, A pincér (the waiter) is the subject of the main clause. Hungarian subjects normally appear in the plain form (nominative) with no ending.

The A is the definite article the. Hungarian uses articles more than some learners expect:

• A pincér = the waiter
• Egy pincér = a waiter

Hungarian normally uses a comma to separate a main clause from a subordinate clause introduced by words like amikor (when), mert (because), hogy (that), etc.

So:
A pincér mosolyog, amikor vizet kérek.
Main clause + comma + time clause.

Yes. Hungarian word order is flexible, and moving the amikor-clause to the front is common to emphasize the timing/condition:

• Amikor vizet kérek, a pincér mosolyog. = When I ask for water, the waiter smiles.

Both are correct; the difference is mostly focus and style.

A common more polite form is conditional:

• …amikor vizet kérnék. = …when I would like to ask for water / when I’d like some water.

Or you can add politeness words:

• …amikor vizet kérek, kérem. (can sound a bit repetitive)
• More natural: …amikor vizet kérek szépen. (when I ask nicely)

In restaurants, you might also say:

• Vizet szeretnék. = I’d like some water. (very common)

Approximate pronunciation (Hungarian is quite consistent):

• pincér: PEEN-tsair (long í like a held ee)
• amikor: AH-mee-kor (stress on the first syllable: Amikor)
• kérek: KAY-rek (long é)
• vizet: VEE-zet (short i, and z like English z)