Наша няня всегда берёт с собой запасной подгузник и маленькое полотенце.

Because няня is grammatically feminine in this meaning (nanny / childminder), so the possessive adjective must also be feminine:

• наш = masculine
• наша = feminine
• наше = neuter
• наши = plural

So:

• наша няня = our nanny

Even though some nouns ending in -я can be masculine in certain cases, няня here is treated as feminine.

Берёт is the 3rd person singular form of the verb брать (to take).

So:

• брать = to take
• он/она берёт = he/she takes

In this sentence, the subject is няня, so:

• Наша няня берёт... = Our nanny takes...

Because the sentence says всегда (always), Russian uses the imperfective verb брать, which fits repeated/habitual actions.

Всегда means always.

Here it comes before the verb:

• Наша няня всегда берёт...

That is very natural. Russian word order is flexible, so you may also hear variations for emphasis, such as:

• Наша няня берёт с собой всегда... — possible, but less neutral
• Всегда наша няня берёт... — more marked/emphatic

The version in the sentence is the most neutral and natural.

С собой means with oneself / along / with her depending on context.

So:

• берёт = takes
• берёт с собой = takes along / brings with her

In this sentence:

• Наша няня всегда берёт с собой... = Our nanny always takes ... with her / brings along ...

The word собой is a form of себя, the reflexive pronoun (oneself).
Russian often uses брать с собой where English might say take along or bring.

They are in the accusative case because they are the direct objects of берёт:

• she takes what?
• запасной подгузник and маленькое полотенце

However, both nouns are inanimate, so in the singular, their accusative forms look like the nominative:

• запасной подгузник (nom.) → запасной подгузник (acc.)
• маленькое полотенце (nom.) → маленькое полотенце (acc.)

That is why they look unchanged.

Because полотенце is a neuter singular noun, and adjectives must agree with the noun in gender, number, and case.

So in nominative/accusative singular:

• masculine: маленький
• feminine: маленькая
• neuter: маленькое

Since полотенце is neuter:

• маленькое полотенце = a small towel

Because the two nouns have different genders:

• подгузник is masculine
• полотенце is neuter

The adjective must match the noun:

• запасной подгузник = masculine
• маленькое полотенце = neuter

So the endings are different because the nouns are different genders, not because the meaning is different.

Yes. Запасной means spare, extra, or backup, depending on context.

Here:

• запасной подгузник = a spare diaper / an extra diaper

It comes from the idea of keeping something in reserve.

Russian has no articles, so there is no direct equivalent of English a, an, or the.

That means:

• запасной подгузник can mean a spare diaper or the spare diaper
• маленькое полотенце can mean a small towel or the small towel

The exact meaning depends on context.

Because и simply joins two objects in a straightforward list:

• запасной подгузник и маленькое полотенце

In Russian, just like in English, you normally do not put a comma before и when it connects two items.

Берёт is pronounced roughly like bee-RYOT.

The letter ё is important because it shows both:

1. the sound yo
2. the stress

So берёт is stressed on the last syllable.

In printed Russian, ё is often written as е, so you may see:

• берет

But in this sentence, the correct pronunciation is still берёт, not берет with a plain e sound.

Usually, not in this context.

• брать / берёт с собой = to take along / bring with oneself
• нести / несёт = to carry, to be carrying

So:

• Наша няня всегда берёт с собой... = she always makes sure to take these things with her
• Наша няня несёт... would focus more on physically carrying them at that moment

For habitual taking along, берёт с собой is the natural choice.

Yes, Russian word order is flexible, but the given order is neutral and natural.

This version sounds like:

• Our nanny always takes with her a spare diaper and a small towel.

You could rearrange parts for emphasis, for example:

• С собой наша няня всегда берёт запасной подгузник и маленькое полотенце.
• Запасной подгузник и маленькое полотенце наша няня всегда берёт с собой.

These are grammatical, but they shift emphasis. The original sentence is the best neutral word order for a learner to model.