मेरी बहन शहर में रहती है, लेकिन माँ गाँव में रहती है।

Because बहन is a feminine noun, and the possessive my must agree with the noun it describes.

• मेरा = my (for masculine singular nouns)
• मेरी = my (for feminine singular nouns)
• मेरे = my (for masculine plural nouns, and also in some oblique contexts)

So:

• मेरी बहन = my sister

If the noun were masculine, you would use मेरा, for example:

• मेरा भाई = my brother

The verb agrees with the subject, and both बहन and माँ are feminine singular nouns.

So:

• रहता है = lives, for a masculine singular subject
• रहती है = lives, for a feminine singular subject

Examples:

• भाई शहर में रहता है। = Brother lives in the city.
• बहन शहर में रहती है। = Sister lives in the city.

In your sentence, both subjects are feminine, so both clauses use रहती है.

में is a postposition meaning in.

So:

• शहर में = in the city
• गाँव में = in the village

Hindi uses postpositions instead of prepositions.
In English, in comes before the noun; in Hindi, में comes after it.

• English: in the city
• Hindi: शहर में

That is just how Hindi grammar works. Hindi uses postpositions, which come after the noun phrase.

A few common examples:

• में = in
• पर = on
• से = from / with / than
• को = to / for / object marker in many cases

So instead of saying something like in city, Hindi says city in:

• शहर में
• गाँव में

This is a very important pattern to get used to in Hindi.

In this sentence, repeating the verb is the most natural and complete way to say it:

• मेरी बहन शहर में रहती है, लेकिन माँ गाँव में रहती है।

Each clause has its own subject and verb:

1. मेरी बहन ... रहती है
2. माँ ... रहती है

In conversation, Hindi can sometimes omit repeated material if the meaning is very clear, but for learners, and in standard complete sentences, repeating रहती है is best.

लेकिन means but.

So the sentence structure is:

• X ..., लेकिन Y ...
  = X ..., but Y ...

Another very common word for but is पर.

For example:

• मैं जाना चाहता हूँ, लेकिन समय नहीं है।
• मैं जाना चाहता हूँ, पर समय नहीं है।

Both are common. लेकिन often feels a bit more formal or explicit, while पर is very common in everyday speech.

Hindi often leaves out a possessive like my when the relationship is already obvious from context.

So:

• मेरी बहन ... लेकिन माँ ... naturally means
• My sister ..., but mother/my mother ...

Since the first clause already tells you whose family is being discussed, माँ is understood as my mother.

You could also say:

• मेरी बहन शहर में रहती है, लेकिन मेरी माँ गाँव में रहती है।

That is also correct. It just sounds a bit more explicit.

In meaning, yes, in this context माँ clearly means my mother.

But grammatically:

• माँ = mother / mom
• मेरी माँ = my mother

Hindi often drops the possessive with close family terms when it is already understood. English can do something similar in limited situations, but Hindi does it more naturally.

The basic Hindi pattern here is:

• Subject + place phrase + verb

So:

• मेरी बहन = subject
• शहर में = place phrase
• रहती है = verb

And then again:

• माँ = subject
• गाँव में = place phrase
• रहती है = verb

This reflects the usual Hindi tendency to put the verb toward the end of the clause.

Here it means lives in the general sense of residence.

The verb comes from रहना, which can mean to stay, to live, or to reside, depending on context.

So:

• वह दिल्ली में रहता है। = He lives in Delhi.
• मैं यहाँ रहता हूँ। = I live here / I stay here.

In your sentence, it refers to where someone lives regularly, so lives is the best translation.

Hindi does not have articles like a, an, and the in the same way English does.

So:

• शहर में can mean in the city or in a city, depending on context
• गाँव में can mean in the village or in a village

In this sentence, English naturally uses the city and the village, but Hindi does not need separate words for that.

Context usually makes the meaning clear.

The postposition में tells you they are being used as locations.

• शहर में = in the city
• गाँव में = in the village

Also, the verb रहना often goes with a place:

• दिल्ली में रहता है = lives in Delhi
• घर में रहती है = lives/stays at home
• गाँव में रहते हैं = live in the village

So the combination of place noun + में + रहना strongly signals location.

Roughly, yes:

• मेरी = my
• बहन = sister
• शहर में = in the city
• रहती है = lives
• लेकिन = but
• माँ = mother / mom
• गाँव में = in the village
• रहती है = lives

A very literal translation would be:

• My sister in the city lives, but mother in the village lives.

That sounds unnatural in English, but it helps show the Hindi structure. A natural English translation is:

• My sister lives in the city, but my mother lives in the village.