Это приказ директора.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Это приказ директора. = This is the director’s order.

But in other tenses, Russian does use a form of быть:

• Это был приказ директора. = This was the director’s order.
• Это будет приказ директора. = This will be the director’s order.

So the missing is is completely normal.

Here это means this in the sense of this is...

It is used to identify something:

• Это приказ директора. = This is the director’s order.

In this kind of sentence, это often works a bit like an identifying word rather than like a regular adjective.

Because это here stands on its own: This is...

Compare:

• Это приказ директора. = This is the director’s order.
• Этот приказ директора = this order of the director / this director’s order

The second example is just a noun phrase, not a complete sentence by itself. So if you want a full sentence meaning This is the director’s order, это is the natural choice.

Приказ is in the nominative singular.

That may feel strange to an English speaker, because in English the noun after is can feel like an object. But in Russian, in a sentence like this, the noun naming what something is is normally in the nominative:

• Это приказ.
• Это книга.
• Это директор.

So приказ is the basic dictionary form here.

Because директора is the genitive singular of директор.

Russian often uses the genitive case to show possession or close relationship between nouns:

• приказ директора = the director’s order
• literally: order of the director

So:

• директор = director
• директора = of the director / director’s

Not in this sentence.

It is true that директора can sometimes be a plural form meaning directors in other contexts. But here, after приказ, the meaning is clearly genitive singular:

• приказ директора = the director’s order

If you wanted the directors’ order, you would usually say:

• приказ директоров

So in this sentence, директора means of the director, not directors.

Yes. That is the most natural way to understand it.

Russian usually does this with case endings instead of an apostrophe:

• English: the director’s order
• Russian: приказ директора

So the ending -а on директора is doing the job that ’s does in English.

Russian has no articles.

So приказ can mean:

• an order
• the order

The exact English choice depends on context. In this sentence, English will often use the:

• This is the director’s order.

But Russian does not mark that difference with a separate word.

Yes, Russian word order is more flexible than English word order.

The version you have:

• Это приказ директора.

is a very normal neutral way to identify something.

Depending on context, you might also hear:

• Приказ директора. = The director’s order.
• Это директора приказ. = possible, but marked or emphatic

For a learner, the given order is the safest and most natural one to use.

A useful stress-marked version is:

• Э́то прика́з дире́ктора.

Approximate pronunciation:

• EH-tuh pree-KAZ dee-RYEK-tuh-ruh

A few notes:

• приказ has stress on the second syllable: прика́з
• директора here has stress on ре: дире́ктора
• Unstressed о is usually reduced in pronunciation

Sometimes yes, depending on context.

The most direct structure is the director’s order or the order of the director, but in natural English that may also come out as:

• This is an order from the director.

What it does not mean is an order for the director. If Russian wanted that idea, it would use a different structure, usually with a preposition.

Приказ usually means an official order, command, or directive given by someone with authority.

So it is stronger and more formal than a casual suggestion. Depending on context, it could be:

• a workplace order
• an administrative directive
• a military command
• an official instruction

So Это приказ директора sounds like something issued by a person in authority, not just a casual request.