Ez a fotó egyedi.

In Hungarian, when a demonstrative like ez (this) comes before a noun, it is normally followed by the definite article too:

• ez a fotó = this photo
• az a ház = that house

So ez a fotó is the standard way to say this photo.
Saying ez fotó would sound incorrect in standard Hungarian.

Word by word:

• ez = this
• a = the
• fotó = photo

But you should understand ez a fotó as one unit meaning this photo, not as a strange literal phrase like this the photo in English. Hungarian simply uses this structure.

Hungarian has two forms of the definite article:

• a before consonant sounds
• az before vowel sounds

Since fotó begins with f, which is a consonant, the correct form is a fotó.

Compare:

• a fotó = the photo
• az alma = the apple

It is not expressed here. In Hungarian, the verb to be is usually omitted in the present tense in the third person singular and plural.

So:

• Ez a fotó egyedi. = This photo is unique.
• literally: This photo unique.

This is normal Hungarian grammar.

But in other persons, to be appears:

• Én egyedi vagyok. = I am unique.
• Te egyedi vagy. = You are unique.

Hungarian adjectives do not agree with nouns in gender, and Hungarian has no grammatical gender at all.

So egyedi stays the same:

• egyedi fotó = a unique photo
• egyedi könyv = a unique book
• egyedi házak = unique houses

Unlike in some languages, the adjective does not change for masculine/feminine/neuter.

Often yes, but it can have a slightly broader meaning than English unique.

egyedi can mean:

• unique
• individual
• distinctive
• one-of-a-kind
• sometimes custom-made or special

So in this sentence, Ez a fotó egyedi, it could mean:

• This photo is unique
• This photo is distinctive
• This photo is special in its own way

The exact nuance depends on context.

Yes. That would mean The photo is unique.

The difference is:

• Ez a fotó egyedi. = This photo is unique.
• A fotó egyedi. = The photo is unique.

So ez a adds the idea of this specific one here.

This sentence uses the most neutral word order:

• Ez a fotó egyedi.

Hungarian word order is flexible, but changes emphasis. For example:

• Ez a fotó egyedi. = neutral statement
• Egyedi ez a fotó. = stronger emphasis on unique

So yes, the words can sometimes move, but the original sentence is the normal, unmarked version.

Approximate pronunciation:

• fotó ≈ foh-TOH
• egyedi ≈ EH-dyeh-dee

A few useful notes:

• Stress in Hungarian is always on the first syllable: FO-tó, E-gye-di
• ó is a long vowel
• gy is a soft consonant sound, somewhat like the dy in duke in some English accents

So the full sentence is roughly:

• EZ a FO-tó E-gye-di

Because egyedi is a predicate adjective here, not a noun.

Structure:

• Ez a fotó = subject
• egyedi = adjective describing the subject

So it works like:

• A ház nagy. = The house is big.
• Ez a fotó egyedi. = This photo is unique.

No article is needed before predicate adjectives.

Yes, sometimes.

• fotó specifically means photo
• kép can mean picture, image, and sometimes photo depending on context

So:

• Ez a fotó egyedi. = specifically This photo is unique.
• Ez a kép egyedi. = This picture/image is unique, or sometimes This photo is unique

If you want to be clearly specific about a photograph, fotó is the better choice.