Mihi hodie epistula brevis scribenda est.

Because mihi is the dative singular of ego, and in this construction Latin uses the dative to show the person who has the obligation.

So mihi ... scribenda est means literally something like:

To me, a short letter must be written today

More natural English is:

I have to write a short letter today

This is a very common Latin pattern with the gerundive + esse.

Scribenda est is a passive periphrastic, a standard Latin way of expressing necessity, obligation, or something that must be done.

It is made of:

• scribenda = the gerundive of scribere (to write)
• est = is

So literally it means:

must be written
or
is to be written

In this sentence, because epistula is the thing that must be written, scribenda agrees with epistula.

Because it agrees with epistula, which is:

• feminine
• singular
• nominative

Latin adjectives, including gerundives, must match the noun they describe in gender, number, and case.

So:

• epistula = feminine singular nominative
• brevis = feminine singular nominative
• scribenda = feminine singular nominative

All three go together: epistula brevis scribenda = the short letter that must be written.

They are both nominative singular.

• epistula is the subject of est
• brevis describes epistula, so it matches it

Even though in English we say I have to write a short letter, Latin structures it differently:

The short letter must be written by me / for me to write

So the letter is grammatically the central noun in the Latin sentence.

Because Latin often prefers this structure when expressing obligation.

Instead of focusing on the person acting, Latin often focuses on the thing that needs to be done:

• Mihi epistula scribenda est
  = I have to write a letter = literally A letter must be written by me

Latin can also express necessity in other ways, but this gerundive + esse + dative pattern is extremely common and important to learn.

Hodie means today.

It is an adverb, so it modifies the whole idea of the action:

Today, I have to write a short letter

Its position is flexible. Latin word order is freer than English word order, so hodie can be placed in different spots for emphasis.

Latin relies much more on endings than on word order to show grammatical relationships.

In this sentence:

• mihi is clearly dative
• epistula is nominative
• brevis matches epistula
• scribenda matches epistula
• est is the verb

Because the endings already show what each word is doing, Latin can move words around more freely. The order here may feel natural or stylistically balanced, but the meaning comes mainly from the forms, not the position.

A very literal translation would be:

To me today a short letter must-be-written is

That sounds unnatural in English, but it helps show the structure.

A better literal-English version is:

A short letter must be written by me today

And the most natural English translation is:

I have to write a short letter today

Because the subject is singular:

• epistula = letter = singular

So Latin uses est = is.

If the subject were plural, the verb would also be plural. For example:

• Mihi hodie epistulae breves scribendae sunt
• I have to write short letters today

Here epistulae, scribendae, and sunt are all plural.

Not exactly, though that is often a useful translation.

In English, if we turn the sentence into passive style, we might say:

A short letter must be written by me

But in Latin, the person with the obligation in this construction is usually put in the dative, not with a/ab.

So mihi is better understood as:

• for me
• by me in sense
• I have to

It marks the person responsible for carrying out the action.

Yes. That would also be a correct way to express the idea:

Ego epistulam scribere debeo = I ought to / must write a letter

But it is a different construction.

• debeo + infinitive focuses more directly on I must
• mihi epistula scribenda est focuses on the letter as something needing to be written

Both are good Latin, but the sentence you were given is teaching the very common passive periphrastic pattern.

Not because of a strict rule. Latin adjective placement is more flexible than English.

Here epistula brevis simply means a short letter. Latin can place adjectives before or after nouns, and the choice may depend on style, rhythm, or emphasis.

What matters most is agreement:

• epistula = feminine singular nominative
• brevis = feminine singular nominative

So they clearly belong together no matter the order.