Перед праздником дети хотят надуть большой шарик.

Because the preposition перед requires the instrumental case.

In this sentence, перед праздником means before the holiday in a time sense. Russian uses:

• перед + instrumental = before / in front of

Examples:

• перед домом = in front of the house
• перед уроком = before class
• перед праздником = before the holiday

So:

• dictionary form: праздник
• after перед: праздником

It can mean both:

• before in time
• in front of in space

So the same preposition is used in different contexts:

• перед праздником = before the holiday
• перед школой = in front of the school

In both meanings, it still takes the instrumental case.

Дети is the normal plural form meaning children.

It is an irregular plural. The singular is usually:

• ребёнок = child

But the plural is:

• дети = children

This is just something you have to learn as a common irregular form.

In Russian, want to do something is usually:

• хотеть + infinitive

So here:

• хотят = they want
• надуть = to blow up / inflate

Together:

• дети хотят надуть = the children want to blow up / inflate

Unlike English, Russian does not use a separate word like to between want and the second verb.

This is a question of aspect.

• надуть = perfective
• надувать = imperfective

Here, надуть is used because the sentence refers to a completed result: the children want to inflate the balloon fully.

Very roughly:

• надувать = to be inflating / to inflate in general / repeatedly
• надуть = to inflate successfully, to blow up

After хотеть, either aspect can be possible depending on meaning, but here the natural choice is надуть because the goal is a finished action.

Here надуть means to blow up or to inflate.

With a balloon, it means filling it with air.

Be careful: in other contexts, надуть can also have figurative meanings, such as to trick or cheat someone in informal Russian. But in this sentence, because of шарик, it clearly means inflate.

Yes, шарик historically is a diminutive form of шар.

• шар = sphere, ball
• шарик = little ball

But in modern Russian, шарик is also a very common everyday word for a balloon, especially a party balloon.

So большой шарик may look like big little-ball, but in real usage it simply means a big balloon. The word шарик is often the normal choice, not necessarily a cute or tiny one.

It is in the accusative case, because it is the direct object of надуть.

The children want to inflate what?

• большой шарик

For inanimate masculine singular nouns, the accusative looks the same as the nominative:

• nominative: большой шарик
• accusative: большой шарик

So even though the form does not change, the function is accusative.

Because it agrees with шарик:

• шарик is masculine
• singular
• inanimate
• here accusative, which matches nominative in this type

So the adjective takes the matching form:

• большой шарик

If the noun were feminine, neuter, or plural, the adjective would change too.

Examples:

• большая коробка
• большое окно
• большие шарики

Yes, Russian word order is fairly flexible.

The neutral order here is:

• Перед праздником дети хотят надуть большой шарик.

This sounds natural and straightforward.

But other orders are possible for emphasis, for example:

• Дети хотят надуть большой шарик перед праздником.
• Большой шарик дети хотят надуть перед праздником.

These versions shift emphasis or sound more marked. The original sentence is a normal, neutral way to say it.

Putting Перед праздником first sets the time frame right away: Before the holiday...

Russian often puts time expressions near the beginning when they give background for the whole sentence.

So this order feels natural:

• Перед праздником = before the holiday
• дети хотят надуть большой шарик = the children want to inflate a big balloon

It is similar to English sentences like:

• Before the holiday, the children want to blow up a big balloon.

Russian has no articles like a or the.

So большой шарик can mean:

• a big balloon
• the big balloon

The exact meaning depends on context.

In an isolated sentence like this, English usually translates it as a big balloon, because that sounds more natural unless the balloon has already been mentioned.

The stress is:

• перед
• пра́здником
• де́ти
• хотя́т
• наду́ть
• большо́й
• ша́рик

A rough pronunciation guide:

• перед = pye-RED
• пра́здником = PRAHZD-ni-kam
• де́ти = DYE-ti
• хотя́т = kha-TYAT
• наду́ть = na-DOOT
• большо́й = bal-SHOY
• ша́рик = SHA-rik

Stress is important in Russian, so it is worth learning each word with its stressed syllable.