Learn Hardcore Hungarian: Ez az óra az asztalon van.

You cannot normally say *ez óra to mean this clock.
Ez óra (without the article) would rather be understood as This is a lesson / hour (and even then it sounds a bit incomplete; more natural: Ez egy óra.).

So in Ez az óra az asztalon van., ez is this, and az is the definite article the. Together ez az óra literally feels like this the clock → this clock.

What is the role of van here, and can it be left out like in other Hungarian sentences?

Van is the 3rd person singular present tense of lenni (to be).
It marks existence or location here: is (located).

In Hungarian, van is often left out in 3rd person singular when:

the predicate is a noun or an adjective:
- Péter orvos. – Péter is a doctor.
- Az óra kerek. – The clock is round.

But with place expressions (answering Where? – hol?), van is normally kept:

Az óra az asztalon van. – The clock is on the table.
Péter otthon van. – Péter is at home.

So:

Ez az óra az asztalon van. is normal.
Ez az óra az asztalon. sounds like a fragment or very colloquial/elliptical, not a full standard sentence.

In short: in location sentences like this, keep van.

Why is it asztalon? What does the ending -on mean?

The ending -on/-en/-ön is the superessive case in Hungarian.
It usually translates as on in English (static location):

asztal – table
asztal
- -on → asztalon – on the table

Other examples:

szék → széken – on the chair
föld → földön – on the ground
pad → padon – on the bench

Which exact form (-on, -en, -ön) you use depends on vowel harmony and pronunciation rules, but the meaning is “on (the surface of)”.

So az asztalon literally means on the table.

Why is the article before asztalon also az, not a?

Hungarian has two forms of the definite article:

a before a consonant
- a könyv – the book
- a szék – the chair
az before a vowel
- az óra – the clock
- az asztal – the table
- az iskola – the school

Since asztal (and asztalon) starts with the vowel a, you must use az:

az asztal – the table
az asztalon – on the table

That’s why the sentence has az asztalon, not a asztalon.

Why is there no egy (like a/an) before óra here? What’s the difference between Ez az óra az asztalon van. and sentences with egy?

az / a = the (definite)
egy = a/an (indefinite)

In Ez az óra az asztalon van., we are talking about a specific, known clock:
ez az óra → this (particular) clock → therefore definite article (az) is used.

If you want to say There is a clock on the table, you would use egy and a different structure:

Van egy óra az asztalon. – There is a clock on the table.
Az asztalon van egy óra. – On the table there is a clock.

Notice:

Ez az óra az asztalon van. – This clock is on the table. (definite)
Van egy óra az asztalon. – There is a clock on the table. (indefinite)

Ez egy óra az asztalon van. is not a correct sentence in standard Hungarian.

Why is the word order Ez az óra az asztalon van and not something like Az óra van az asztalon or Ez az óra van az asztalon?

Hungarian word order mainly expresses topic and emphasis (focus), not tense or subject-verb agreement.

A neutral answer to the question Hol van ez az óra? (Where is this clock?) is:

Ez az óra az asztalon van.

Rough structure:

Ez az óra – topic (what we’re talking about)
az asztalon – focused information (where it is)
van – verb

Variants:

Az óra az asztalon van. – The clock is on the table. (no ez; still neutral)
Az asztalon van ez az óra. – now az asztalon is in a very prominent position; often used when you contrast it with another place (It’s on the table, not on the shelf).
Ez az óra van az asztalon. – often stresses which thing is on the table (e.g. It’s this clock that is on the table, not that one).

So Ez az óra az asztalon van. is the normal, neutral way to state where this clock is.

What exactly does óra mean here? It can mean clock, hour, or lesson, right?

Yes, óra is polysemous in Hungarian. It can mean:

clock / watch
- falióra – wall clock
- karóra – wristwatch
hour (unit of time)
- két óra – two hours / two o’clock (depending on context)
lesson / class period
- matekóra – math lesson / math class

In Ez az óra az asztalon van., the only natural interpretation is clock / watch, because an hour or a lesson being “on the table” doesn’t make sense physically.

Context decides the meaning. If you need to be explicit:

Ez a falióra az asztalon van. – This wall clock is on the table.
Ez a matekóra húsz perc múlva kezdődik. – This math class starts in 20 minutes.

How would I say onto the table or from the table, and how is that different from az asztalon?

Hungarian uses different case endings for on / onto / from:

on the table (static position) → az asztalon
- Ez az óra az asztalon van. – This clock is on the table.
onto the table (movement to the surface) → az asztalra
- Az órát az asztalra teszem. – I’m putting the clock onto the table.
from (off) the table → az asztalról
- Az órát az asztalról veszem le. – I’m taking the clock off the table.

So:

asztalon – on (where?)
asztalra – onto (to where?)
asztalról – from (off) (from where?)

How do you make this sentence plural, like These clocks are on the table?

You pluralize both the demonstrative and the noun, and use the plural form of van:

Ez az óra az asztalon van. – This clock is on the table.
Ezek az órák az asztalon vannak. – These clocks are on the table.

Breakdown:

ez → ezek (this → these)
óra → órák (clock → clocks)
van → vannak (is → are)

The article az stays the same before órák, because it still starts with a vowel.

How do you say this sentence in the negative, like This clock is not on the table?

Hungarian uses nem (not) and a special negative form of van: nincs.

Singular:

Ez az óra az asztalon van. – This clock is on the table.
Ez az óra nincs az asztalon. – This clock is not on the table.

Plural:

Ezek az órák az asztalon vannak. – These clocks are on the table.
Ezek az órák nincsenek az asztalon. – These clocks are not on the table.

So in the negative, instead of nem van, you normally use nincs / nincsenek.

Can I also say Ez az óra itt van az asztalon? What is the difference?

Yes:

Ez az óra az asztalon van. – This clock is on the table.
Ez az óra itt van az asztalon. – roughly: This clock is right here on the table.

itt = here, ott = there.

Adding itt or ott:

makes the location more obvious / pointed out:
- Ez az óra itt van az asztalon. – It’s here on the table (not anywhere else).
- Ez az óra ott van az asztalon. – It’s there on the table (over there).

Without itt/ott, the sentence just neutrally states the location. With itt/ott, you are usually gesturing or contrasting it with another place.