Για την αίτηση χρειάζομαι ένα έγγραφο από τη γραμματεία και δύο φωτοτυπίες της ταυτότητάς μου.

It is a prepositional phrase that sets the context: for the application.

Greek often puts this kind of phrase first to frame the whole sentence. So the sentence is not just I need... but more specifically For the application, I need...

You could also put it later: Χρειάζομαι ένα έγγραφο ... για την αίτηση but the original order sounds very natural.

Because για is followed by the accusative case.

The basic dictionary form is:

• η αίτηση = the application

After για, it becomes:

• την αίτηση = for the application

So this is a case change:

• nominative: η αίτηση
• accusative: την αίτηση

Because Greek uses the definite article very often, and here it refers to a specific application, not just an application in general.

So:

• για αίτηση would sound more general or less specific
• για την αίτηση means for the application / for this application

Χρειάζομαι means I need.

It is:

• 1st person singular
• present tense

A useful thing to notice is that it ends in -μαι, which looks like a passive/middle form, but its meaning here is active: I need. Verbs like this are very common in Modern Greek.

In this sentence, χρειάζομαι takes direct objects:

• ένα έγγραφο
• δύο φωτοτυπίες

Because έγγραφο is a neuter singular noun.

So the indefinite article must match it:

• masculine: ένας
• feminine: μια / μία
• neuter: ένα

That is why Greek says:

• ένα έγγραφο = a document

In Modern Greek, από normally takes the accusative.

So:

• η γραμματεία = the office / secretariat (nominative)
• τη(ν) γραμματεία = the office / secretariat (accusative)

That is why the sentence has:

• από τη γραμματεία = from the office/secretariat

This is worth noticing because English speakers sometimes expect something like a genitive after from, but Modern Greek usually uses the accusative after prepositions like από.

Both are forms of the same article: την.

In normal Modern Greek spelling and pronunciation, the final -ν is often dropped before certain consonants. Since γραμματεία begins with γ, you usually get:

• τη γραμματεία

So:

• την αίτηση
• τη γραμματεία

This is completely normal and not a different article.

Because δύο means two, so the noun goes in the plural:

• μία φωτοτυπία = one photocopy
• δύο φωτοτυπίες = two photocopies

Also, there is no article here because it is an indefinite quantity:

• δύο φωτοτυπίες = two photocopies not
• οι δύο φωτοτυπίες = the two photocopies

Because this phrase depends on φωτοτυπίες and means of my ID.

Greek uses the genitive for this kind of relationship:

• φωτοτυπίες της ταυτότητας = photocopies of the ID

The basic noun is:

• η ταυτότητα = the ID / identity card

In the genitive singular it becomes:

• της ταυτότητας

With μου, the written form becomes:

• της ταυτότητάς μου

So the whole phrase means:

• two photocopies of my ID

Yes, μου means my, but Greek normally still keeps the article.

So Greek says:

• η ταυτότητά μου = my ID
• της ταυτότητάς μου = of my ID

This is very normal Greek structure. In English, you usually do not say of the my ID, but Greek grammar works differently: the article and the possessive can appear together.

Because the weak possessive forms such as μου, σου, του, της usually come after the noun in Greek.

For example:

• το σπίτι μου = my house
• η αίτησή μου = my application
• η ταυτότητά μου = my ID

So μου is postposed, not placed before the noun the way English my is.

This happens because μου is an enclitic. In Greek, when certain unstressed words like μου follow a word, the previous word may receive an additional written accent.

That is why you see forms like:

• η ταυτότητά μου
• της ταυτότητάς μου

This is a spelling-and-stress rule, and it is very common with possessive μου, σου, etc.

Yes. Greek word order is more flexible than English word order.

For example, you could also say:

• Χρειάζομαι για την αίτηση ένα έγγραφο από τη γραμματεία και δύο φωτοτυπίες της ταυτότητάς μου.

But the original:

• Για την αίτηση χρειάζομαι ...

sounds very natural because it puts the purpose first and tells the listener immediately what this is about.