Learn Hardcore Hungarian: A képük a falon van.

It is the third-person plural possessive suffix: -uk/-ük, meaning their.
Because kép has a front vowel (é), it takes the front-harmony form -ük: kép + -ük → képük = their picture (one item).
Note: after a consonant you use -uk/-ük, not -juk/-jük, so képjük is incorrect.

Why is there a definite article A before a possessed noun: A képük?

Hungarian normally uses the definite article (a/az) with possessed nouns. Even though English doesn’t say “the their picture,” Hungarian treats possessed nouns as definite. So you say A képük (their picture), A házunk (our house), etc. Omitting the article changes the nuance (often making it existential/indefinite or sounding elliptical).

Why do we need van here? I thought Hungarian often drops “to be” in the present tense.

Hungarian omits the copula in 3rd person present with nominal or adjectival predicates (for example: A képük szép = Their picture is pretty). But you must use van/vannak when:

indicating location, time, or existence: A képük a falon van (Their picture is on the wall).
forming existential sentences: Van egy képük (They have a picture / There is a picture of theirs).

Why is it van and not vannak?

The verb agrees with the grammatical subject. Képük is singular (one picture), so van. If the possessed item is plural, you use vannak:

Singular possessed: A képük a falon van (Their picture is on the wall).
Plural possessed: A képeik a falon vannak (Their pictures are on the wall).

What does the ending -on in falon mean?

It is the superessive case, meaning on. Hungarian uses case endings instead of prepositions for many locations:

-on/-en/-ön = on (choice depends on vowel harmony and word shape)
fal + -on → falon = on the wall Compare: asztalon (on the table), képernyőn (on the screen).

Could I change the word order, e.g., A falon van a képük?

Yes, and it changes the emphasis:

A képük a falon van. Neutral: topic = their picture; comment = it is on the wall.
A falon van a képük. Focus on location: it’s on the wall (as opposed to elsewhere).
A képük van a falon. Focus on identity: it’s their picture (as opposed to someone else’s).

How do I say “Their pictures are on the wall”?

Use the plural possessed form and plural verb:

A képeik a falon vannak. Notes:
Plain plural: képek = pictures.
Plural when possessed: the plural appears as -i- before the possessive: képeik (their pictures).

How do I say “One of their pictures is on the wall”?

Az egyik képük a falon van. For several: Néhány képük a falon van (Some of their pictures are on the wall).

Does képük mean “a picture belonging to them” or “a picture of them”?

On its own, képük most naturally means a picture that they own or that is associated with them. If you specifically mean a picture depicting them, say:

kép róluk (a picture of them)
a róluk készült kép (the picture taken of them) Context often disambiguates, but the possessive alone can be ambiguous.

Can I omit the first article and say Képük a falon van?

In a neutral declarative sentence, that sounds off. Possessed nouns typically take the definite article when used as subject or object. You can drop the article in certain structures (especially existential or with quantifiers), e.g., Van képük a falon (They have a picture on the wall), but for “Their picture is on the wall,” use A képük.

How do I negate the sentence?

Use nincs/nincsenek with location:

A képük nincs a falon. (Their picture is not on the wall.)
Plural: A képeik nincsenek a falon. Do not say nem van in this kind of sentence.

Is there a more idiomatic verb than van for a picture on a wall?

Yes, lóg (hangs) is common:

A képük a falon lóg. (Their picture is hanging on the wall.) This emphasizes the manner (hanging) rather than mere location.

How do I explicitly say “their” with a separate word?

Add the emphatic pronoun az ő:

Az ő képük a falon van. (It’s their picture that is on the wall.) This stresses ownership (theirs, not someone else’s).

How do I ask “Where is their picture?”

Hol van a képük? Answer: A falon van. Plural possessed: Hol vannak a képeik? — A falon vannak.

When do I use a vs. az?

Both are the definite article. Use a before consonants and az before vowels:

a falon (on the wall)
az asztalon (on the table)
az ő képük (their picture) because ő starts with a vowel.