Learn Hardcore Croatian: Moramo napraviti prijavu za tečaj hrvatskog jezika. - We have to register for the Croatian language course.

Moramo = we must / we have to / we need to.
It expresses an obligation or necessity, just like we have to in everyday English.
Strength-wise, it can cover both strong obligation (we must) and practical necessity (we need to / we have to), depending on context and tone.

You cannot say just Moramo prijavu...; moramo must be followed by an infinitive verb (here: napraviti).

Why is napraviti used, and what is the difference between napraviti and something like raditi or učiniti?

Napraviti is a perfective verb meaning to make / to do (and complete it).

napraviti prijavu ≈ to make an application / to file an application (and get it done).
It focuses on completing the action (one finished application).

Compare:

raditi prijavu – to work on an application, focusing on the process, not on completion.
učiniti – also to do / to make, but here napraviti is the more natural collocation with prijava.

In this sentence, moramo napraviti implies that the goal is to complete the application, not just work on it.

Why is it prijavu and not prijava?

Prijava is a feminine noun meaning application / registration / report.

prijava – nominative singular (dictionary form)
prijavu – accusative singular

In the sentence:

napraviti prijavu – to make an application

The verb napraviti takes a direct object in the accusative, so prijava changes to prijavu.

So:

Prijava je gotova. – The application is finished. (subject → nominative)
Moramo napraviti prijavu. – We have to make an application. (object → accusative)

Could I say Moramo se prijaviti na tečaj hrvatskog jezika instead? Is that more natural?

Yes, and many speakers would actually prefer that version in everyday speech:

Moramo se prijaviti na tečaj hrvatskog jezika.
≈ We have to register for the Croatian language course.

Differences:

Moramo napraviti prijavu za tečaj...
- More literal: We have to make an application for the course.
- Sounds a bit more bureaucratic / formal, focusing on the application document or procedure.
Moramo se prijaviti na tečaj...
- Uses the reflexive verb prijaviti se = to register / sign up.
- Feels more direct and colloquial, focusing on the act of registering.

Both are correct; choice depends on style and what you want to emphasize.

Why is the preposition za used with tečaj? Could I use na instead?

In the sentence:

prijavu za tečaj hrvatskog jezika
- za
  - accusative (tečaj) = for the course

za here expresses purpose or target: an application for something.

If you use the reflexive verb:

prijaviti se na tečaj – to register for a course
- Here, na
  - accusative means something like onto / to (an event, course, etc.).

So:

prijava za tečaj – an application for a course
prijaviti se na tečaj – to sign up for a course

za and na are both correct but used with different constructions:

za with the noun prijava
na with the verb prijaviti se

Why is it hrvatskog jezika and not just hrvatski jezik or hrvatski?

Hrvatski jezik = Croatian language (nominative).
In tečaj hrvatskog jezika, we have a genitive construction:

tečaj (čega?) hrvatskog jezika – course of Croatian language

So:

hrvatskog – genitive masculine singular of the adjective hrvatski
jezika – genitive singular of jezik (language, tongue)

This genitive (X of Y) is the usual way to say course of [language] in Croatian.

You would say:

tečaj engleskog jezika – English language course
tečaj njemačkog jezika – German language course

Using hrvatski jezik in nominative here (tečaj hrvatski jezik) would be ungrammatical in standard Croatian.

Can I drop jezika and just say tečaj hrvatskog?

Yes, in context it is very common and natural to say:

tečaj hrvatskog – Croatian course (literally: course of Croatian)

jezika is understood from context and often omitted, especially in informal or semi-formal speech.

More formal and explicit:

tečaj hrvatskog jezika

More concise but still correct:

tečaj hrvatskog

Could I say Moramo prijaviti za tečaj hrvatskog jezika without napraviti?

No, that would be wrong in standard Croatian.

Options:

With a noun:
- Moramo napraviti prijavu za tečaj hrvatskog jezika.
  (make an application)
With a reflexive verb:
- Moramo se prijaviti na tečaj hrvatskog jezika.
  (register / sign up)

The verb prijaviti without se usually means to report someone (to the police, authorities, etc.), not to apply for a course. So:

prijaviti se – to register oneself
napraviti prijavu – to make an application
prijaviti nekoga – to report someone

You must either:

keep napraviti with the noun prijavu, or
use the reflexive verb prijaviti se with na tečaj.

What is the role of verb aspect here? Why napraviti (perfective) and not praviti (imperfective)?

Croatian has aspect: perfective vs imperfective.

napraviti – perfective: to make / to do (and finish it)
praviti – imperfective: to be making / doing (in progress, repeated, or habitual)

In this sentence:

Moramo napraviti prijavu... – we have an obligation to complete the application. The focus is on one finished action.

If you said:

Moramo praviti prijavu...

it would sound odd; it would emphasize the process of making the application (as an ongoing activity), not the finished result. For tasks to be completed, the perfective (napraviti) is normally used.

How flexible is the word order? Can I move parts of the sentence around?

Word order in Croatian is relatively flexible because of cases, but the neutral version here is:

Moramo napraviti prijavu za tečaj hrvatskog jezika.

You can move parts for emphasis:

Za tečaj hrvatskog jezika moramo napraviti prijavu.
– Emphasis on for the Croatian course.
Prijavu za tečaj hrvatskog jezika moramo napraviti.
– Emphasis on the application itself.

All these are grammatically correct. The standard, most neutral spoken form is the one you were given.

Is there any difference in tone if I use trebamo instead of moramo?

Yes, there is a small nuance:

moramo – we must / we have to, often feels a bit stronger, more obligatory.
trebamo – we should / we need to, often sounds slightly softer or more like a recommendation.

So:

Moramo napraviti prijavu za tečaj hrvatskog jezika.
– Stronger obligation.
Trebamo napraviti prijavu za tečaj hrvatskog jezika.
– Still an obligation, but can sound a bit less strict, depending on context.

Both are very common and grammatically correct.