Нужно подать заявление.

Нужно is an impersonal predicative word meaning it is necessary / one must / it’s needed.

So:

• Нужно подать заявление. = It is necessary to submit an application.

There is no explicit subject here. The sentence does not say exactly who has to do it. It just states the necessity in a general way.

Compare:

• Мне нужно подать заявление. = I need to submit an application.
• Вы должны подать заявление. = You must submit an application.

A form with должен / должна / должны is more directly tied to a person and agrees with that person in gender/number. Нужно is more neutral and often sounds like an instruction, requirement, or general necessity.

Because after нужно, Russian normally uses the infinitive to express what action is necessary.

Pattern:

• нужно + infinitive

Examples:

• Нужно подождать. = It’s necessary to wait.
• Нужно заполнить форму. = It’s necessary to fill out the form.
• Нужно подать заявление. = It’s necessary to submit an application.

So подать is not being conjugated because the sentence is built around the idea it is necessary to do X.

This is a question of aspect.

• подать = perfective
• подавать = imperfective

Here, подать заявление means to submit an application as a complete, one-time action. That is why perfective подать is the natural choice.

Use подавать when you mean:

• repeated action,
• process,
• general habit,
• or the action without focusing on completion.

Compare:

• Нужно подать заявление до пятницы.
  You need to submit the application by Friday.
  One completed submission.

• Обычно заявления нужно подавать заранее.
  Applications usually need to be submitted in advance.
  General rule/habit, so imperfective.

Заявление is in the accusative case here, because it is the direct object of подать.

In this specific noun, the accusative looks the same as the nominative:

• nominative: заявление
• accusative: заявление

This happens because заявление is:

• neuter
• inanimate
• singular

For many inanimate nouns in Russian, the accusative singular is identical to the nominative.

Заявление often means a formal application, statement, or written request, depending on context.

In bureaucratic or official contexts, it very often means:

• application
• formal request
• written petition/request

So in a sentence like this, it usually refers to an official document you submit to an institution, employer, university, government office, etc.

Common collocations include:

• подать заявление = submit an application
• написать заявление = write an application/request
• подписать заявление = sign an application

By itself, Нужно подать заявление. is general and impersonal. It does not specify who must do it.

Depending on context, English might translate it as:

• It is necessary to submit an application.
• You need to submit an application.
• One must submit an application.
• An application needs to be submitted.

Russian often leaves the responsible person unstated when it is obvious from context or not important.

If you want to name the person, you can add a dative noun or pronoun:

• Мне нужно подать заявление. = I need to submit an application.
• Вам нужно подать заявление. = You need to submit an application.
• Студенту нужно подать заявление. = The student needs to submit an application.

Because Russian very often uses subjectless impersonal sentences to talk about necessity, possibility, weather, states, and similar ideas.

In English, we often insert a subject like it:

• It is necessary to...

In Russian, that dummy subject is usually not needed:

• Нужно подать заявление.

This is completely natural Russian. The sentence is grammatically complete even without a named subject.

Yes. Russian word order is flexible, although the neutral order here is very natural:

• Нужно подать заявление.

You may also see:

• Заявление нужно подать.

This version puts more emphasis on заявление—as if contrasting it with something else:

• Заявление нужно подать сегодня, а документы — завтра.
  The application needs to be submitted today, and the documents tomorrow.

So both are correct, but the word order can slightly change emphasis.

Yes. Надо подать заявление is very close in meaning.

Both надо and нужно can mean:

• need to
• must
• it is necessary to

Very roughly:

• надо often sounds a little more conversational/common
• нужно can sound slightly more neutral or formal

But in many real situations they are interchangeable.

Compare:

• Нужно подать заявление.
• Надо подать заявление.

Both are natural.

Add the person in the dative case:

• Мне нужно подать заявление.

Why dative? Because with нужно, the person who experiences the necessity is often marked in the dative:

• мне = to me / for me
• вам = to you / for you
• ему = to him
• ей = to her
• нам = to us
• им = to them

Examples:

• Мне нужно подать заявление. = I need to submit an application.
• Вам нужно подать заявление сегодня. = You need to submit an application today.

It is generally neutral, and it fits especially well in official, administrative, or instructional contexts.

Because of the words:

• подать = submit/file
• заявление = formal application/request

the whole sentence naturally sounds a bit bureaucratic or official, even though the grammar itself is not especially stiff.

You might hear it:

• at a government office,
• in university administration,
• in HR,
• in written instructions,
• in formal conversation.

So the grammar is neutral, but the vocabulary makes it feel fairly official.