Мне нужно купить новую зубную щётку.

Because Russian often expresses need with an impersonal structure:

Мне нужно купить новую зубную щётку.
Literally: To me, it is necessary to buy a new toothbrush.

So:

• я = I (subject form)
• мне = to me / for me (dative form)

In English, we say I need to buy...
In Russian, a very common pattern is [dative pronoun] + нужно + infinitive.

Examples:

• Мне нужно идти. = I need to go.
• Тебе нужно отдохнуть. = You need to rest.
• Ему нужно позвонить. = He needs to call.

So мне is used because Russian is not saying I need, but something more like it is necessary for me.

Нужно means necessary, needed, or need to, depending on context.

In this sentence, нужно works as an impersonal predicate:

• Мне нужно купить... = I need to buy...
• literally: It is necessary for me to buy...

It does not change for person:

• Мне нужно
• Тебе нужно
• Ему нужно
• Нам нужно

This makes it different from English need, which changes with the subject only a little, and different from Russian verbs that fully conjugate.

A close synonym is надо:

• Мне нужно купить новую зубную щётку.
• Мне надо купить новую зубную щётку.

Both are common. Надо is often a bit more conversational.

Because купить is the perfective form, and here it means to buy / to get one completed purchase done.

Russian has aspect:

• покупать = imperfective, process / repeated action / general activity
• купить = perfective, one completed action

In this sentence, the speaker means they need to go and buy a toothbrush, as a single completed action. So купить is the natural choice.

Compare:

• Мне нужно купить новую зубную щётку.
  = I need to buy a new toothbrush.
  (one purchase, completed result)

• Мне нужно покупать новые зубные щётки чаще.
  = I need to buy new toothbrushes more often.
  (habit / repeated action)

So купить fits because the sentence is about a single concrete purchase.

Because it is the direct object of купить, so it goes into the accusative case.

Dictionary form:

• щётка = toothbrush / brush (nominative singular)

In the sentence:

• купить щётку = to buy a toothbrush

So:

• nominative: щётка
• accusative: щётку

This is normal for many feminine nouns ending in -а / -я:

• книга → книгу
• машина → машину
• лампа → лампу
• щётка → щётку

The noun changes because Russian marks grammatical role with case endings.

Because adjectives must agree with the noun they describe in:

• gender
• number
• case

The noun щётка is:

• feminine
• singular
• accusative

So both adjectives must also be feminine singular accusative:

• новая → новую
• зубная → зубную

Full phrase:

• nominative: новая зубная щётка
• accusative: новую зубную щётку

This agreement is one of the most important patterns in Russian grammar.

Yes. Зубная щётка is the normal standard way to say toothbrush.

Literally:

• зуб = tooth
• зубной / зубная / зубное = dental, tooth-
• щётка = brush

So зубная щётка literally means tooth brush or dental brush, but in normal English translation it is simply toothbrush.

This is a fixed everyday expression.
You may also understand something like щётка для зубов (brush for teeth), but зубная щётка is the usual phrase.

Because Russian has no articles.

English:

• a new toothbrush
• the new toothbrush

Russian:

• новую зубную щётку

Russian does not have separate words like a/an/the.
Whether something is definite or indefinite is understood from context.

So новую зубную щётку could mean:

• a new toothbrush
• the new toothbrush

depending on the situation. In this sentence, English would normally translate it as a new toothbrush.

Russian word order is more flexible than English, although some orders sound more neutral than others.

The most neutral version is:

Мне нужно купить новую зубную щётку.

But other orders are possible, depending on emphasis:

• Новую зубную щётку мне нужно купить.
  Emphasizes the new toothbrush.

• Купить новую зубную щётку мне нужно.
  Emphasizes the action to buy.

• Мне новую зубную щётку нужно купить.
  Also possible; focus may fall on the object.

A learner should usually start with the neutral order: мне нужно купить новую зубную щётку.

A few useful pronunciation points:

• ё is pronounced like yo
• щ is a soft sound, often approximated for learners as something between sh and shch
• ё is always stressed

So щётка sounds roughly like SHYOT-ka or shchyot-ka, depending on how detailed you want to be.

Important note: in many Russian texts, ё is often written as е, especially informally or in print. So you may see:

• щётка
• щетка

But the pronunciation is still щётка, not щетка with a plain e sound.

So if you see зубную щетку, it usually still means зубную щётку.

Yes, but it means something slightly different.

• Мне нужно купить новую зубную щётку.
  = I need to buy a new toothbrush.
  Neutral statement of necessity.

• Я должен купить новую зубную щётку.
  = I must / I am supposed to buy a new toothbrush.
  Stronger sense of obligation, duty, or requirement.

So нужно is often the more natural choice for everyday practical needs.
Должен can sound stronger, more like an obligation.

Compare:

• Мне нужно купить молоко. = I need to buy milk.
• Я должен купить молоко. = I have to buy milk / I’m obliged to buy milk.

The main stressed syllables are:

• мне
• ну́жно
• купи́ть
• но́вую
• зубну́ю
• щё́тку

So the sentence is:

Мне ну́жно купи́ть но́вую зубну́ю щё́тку.

A useful thing to remember:

• ё is always stressed, so щётку is easy to stress correctly.
• Stress in Russian is important because it is not always predictable.

If you learn the sentence as a chunk, it becomes easier: Мне ну́жно купи́ть но́вую зубну́ю щё́тку.