A megállapodást ma aláírjuk.

Because it is the direct object of the verb aláírjuk.

Hungarian usually marks a direct object with the accusative ending -t:

• megállapodás = agreement
• megállapodást = agreement as the thing being signed

So A megállapodást ma aláírjuk means that the agreement is what we will sign.

Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since megállapodást starts with m, the correct article is a.

So:

• a megállapodás
• a megállapodást

Because Hungarian has two kinds of verb conjugation: indefinite and definite.

You use the definite conjugation when the verb has a definite object, such as:

• a megállapodást = the agreement

So:

• aláírjuk = we sign it / we will sign it
• aláírunk would be used when there is no definite object, or the object is indefinite

That is why aláírjuk is the correct form here.

The verb form is present tense, but the sentence can still refer to the future.

Hungarian often uses the present tense for a planned or scheduled future action, especially when there is a time word like ma.

So A megállapodást ma aláírjuk means something like:

• We are signing the agreement today
• We will sign the agreement today

If you want to make the future more explicit, Hungarian can also say:

• Ma alá fogjuk írni a megállapodást.

Because the verb ending already tells you the subject.

In aláírjuk, the ending -juk shows:

• 1st person plural = we
• and also that it is the definite conjugation

So Hungarian usually leaves out the subject pronoun unless it is needed for emphasis:

• Mi aláírjuk. = We will sign it.
  This sounds emphatic, like we (not someone else).

Yes. Hungarian word order is quite flexible, and different orders can sound more neutral or put emphasis in different places.

For example:

• A megállapodást ma aláírjuk.
• Ma aláírjuk a megállapodást.

Both are natural.

The version with A megállapodást first makes the agreement the starting point or topic of the sentence, something like:

• As for the agreement, we’ll sign it today.

The version Ma aláírjuk a megállapodást is often felt as a more neutral all-purpose order.

It comes from:

• alá = under, beneath
• ír = write

So aláír literally means write under, which is how Hungarian says sign.

This is very common in Hungarian: many verbs are built from a basic verb plus a prefix.

Yes, the prefix can separate in some sentence patterns.

In this sentence, it stays attached:

• aláírjuk

But with negation, focus, some questions, and some other structures, the prefix often moves after the verb:

• Nem írjuk alá. = We are not signing it.
• Mikor írjuk alá? = When are we signing it?
• Ma írjuk alá. = It is today that we are signing it.

So the attached form aláírjuk is normal here, but Hungarian prefix verbs do not always stay together.