Learn Hardcore Hungarian: A lány a padon ülve telefonál. - The girl is making a phone call while sitting on the bench.

So a padon ülve literally means “being-on-the-bench sitting”, i.e. “(while) sitting on the bench”.
It tells you how / in what position / under what circumstance the girl is making the phone call.

How is ülve formed from ül?

You take the verb stem and add -va or -ve:

If the stem ends in a front vowel word and the consonant cluster allows it, you usually add -ve.
If not, you add -va.

Here: ül → ülve (sit → sitting).

This pattern works for many verbs:

áll → állva (stand → standing)
fekszik → fekve (lie → lying)
mosolyog → mosolyogva (smile → smiling)

Why is there no separate word for “while” in this sentence?

Hungarian often uses the -va/-ve participle instead of a separate word like while.

A lány a padon ülve telefonál.
= The girl is phoning while (she is) sitting on the bench.

The -va/-ve ending itself carries the idea of simultaneity (or sometimes sequence), so you don’t need a word like while or as. The ülve form already encodes that the sitting and phoning happen at the same time.

What is the difference between ül and ülve in meaning and function?

ül is a finite verb: it has person and number, and can be the main verb of the clause.
- A lány a padon ül. – The girl is sitting on the bench.
ülve is a non-finite form (adverbial participle): it cannot be the main verb by itself. It modifies another verb and describes how / in what state / when that main action happens.
- A lány a padon ülve telefonál. – The girl is phoning (while) sitting on the bench.

So ülve behaves more like English “sitting” in “sitting on the bench, she’s phoning” than like “sits”.

Does ülve agree with the subject in person, number, or gender?

No. The -va/-ve participle never changes for person, number, or gender. It has one form only.

A lány a padon ülve telefonál. – The girl…
A fiúk a padon ülve telefonálnak. – The boys…

In both cases, ülve stays the same. Only the main finite verb (telefonál / telefonálnak) changes.

Could I instead say A lány a padon ül és telefonál? Is that the same?

Yes, you can say:

A lány a padon ül és telefonál.

This is perfectly correct and very natural. The nuance:

ül és telefonál: simply lists two actions: she sits, and she phones.
ülve telefonál: treats the sitting as a background state or manner for the main action (phoning) – like phoning while sitting, phoning in a sitting position.

In most everyday contexts, the meaning is very close, and both are fine.

Why is it a padon and not some other form like a padra?

In Hungarian, -on / -en / -ön (here: -on) is the superessive case, meaning “on (the surface of)”.

pad – bench
a padon – on the bench

Other cases would change the meaning:

a padra – onto the bench (movement to the bench)
a padnál – by / near the bench

Since the girl is located on the bench while she phones, a padon is the correct form.

Why do we have the article a twice: A lány a padon?

Hungarian uses the definite article a / az before each definite noun phrase, even if English would only use it once.

A lány a padon ülve telefonál.
Literally: “The girl on the bench sitting phones.”

Both lány and pad are definite, so each gets its own article:

a lány – the girl
a padon – on the (specific) bench

This is normal and required in Hungarian.

Is the word order fixed, or can I change it, like A lány telefonál a padon ülve?

Hungarian word order is flexible, but information structure and emphasis matter.

A lány a padon ülve telefonál.
Neutral: tells where/how she is while phoning.

Other possibilities:

A lány telefonál a padon ülve. – Grammatically possible, but sounds a bit odd; it can feel like you’re tacking on on the bench, sitting as afterthought information.
A padon ülve telefonál a lány. – Emphasises “sitting on the bench” (not somewhere else, or not in some other posture).

In practice, A lány a padon ülve telefonál is the most natural neutral version.

Why do we say telefonál and not something like telefonálni here?

Telefonál is the finite verb in the sentence:

telefonál – he/she phones / is phoning (3rd person singular, present)

Telefonálni is the infinitive (to phone), which cannot serve as the main verb of the clause by itself.

Since we need a main verb for what the girl is doing, we must use telefonál, not telefonálni.

How does Hungarian express “is phoning” if there’s no special progressive tense?

Hungarian usually uses the simple present for both:

A lány telefonál.
= The girl phones. or The girl is phoning.

Context, adverbs, and situation tell you whether it’s a general habit or an ongoing action.
In A lány a padon ülve telefonál, the added circumstance a padon ülve strongly suggests an ongoing action right now (she’s currently phoning while sitting on the bench).

Could I rephrase this with a miközben (“while”) clause? How would that compare?

Yes, you could say:

A lány a padon ül, miközben telefonál.
The girl is sitting on the bench while she is phoning.

A lány telefonál, miközben a padon ül.

Using miközben is closer to English “while”. The -va/-ve form (ülve) is usually more compact and somewhat more written or formal, but it’s also very common in speech.
In meaning, they are essentially the same here: simultaneous actions.

Can ülve be used with other verbs in the same way, or is this special to ül?

The pattern is completely general. You can use many verbs in the -va/-ve form to describe how another action happens:

A fiú állva beszél. – The boy is speaking while standing.
Mosolyogva köszönt. – She greeted (him) with a smile / smiling.
Futva érkezett. – He arrived running.

In your sentence, ülve just happens to be the -va/-ve form of ül, but the construction itself is widely used.