Mój brat nie chce być fryzjerem ani dentystą; woli pracować w metrze przy nowym tunelu.

Because mój has to agree with brat in gender, number, and case.

• brat is masculine singular
• so the correct form is mój

Compare:

• mój brat = my brother
• moja siostra = my sister
• moje dziecko = my child

So this is standard adjective/possessive agreement.

Polish normally does not use articles.

So where English says:

• my brother
• a hairdresser
• the subway

Polish just says:

• mój brat
• fryzjer
• metro

Whether something is a or the is usually understood from context. This is one of the biggest differences from English.

Because chce is the 3rd person singular form of chcieć = to want.

The subject is mój brat = my brother, so the verb must mean he wants:

• chcę = I want
• chcesz = you want
• chce = he/she/it wants

So mój brat nie chce... means my brother does not want...

Because Polish often omits subject pronouns when the subject is already clear.

Here, mój brat already tells you who the subject is, so adding on would usually be unnecessary.

• Mój brat nie chce... = natural
• On nie chce... = possible, but more emphatic or contrastive

Polish does this much more than English.

With verbs, nie is usually written separately.

So:

• nie chce = does not want
• nie pracuje = does not work
• nie był = was not

That is the normal rule for verbal negation in Polish.

Because after verbs like chcieć = to want, Polish usually uses an infinitive to say what someone wants to do or be.

So:

• chce być = wants to be
• chce pracować = wants to work
• woli pracować = prefers to work

This works very similarly to English want to be and prefer to work.

Because after być in this kind of sentence, Polish normally uses the instrumental case for professions, roles, and identities.

Basic forms:

• fryzjer = hairdresser / barber
• dentysta = dentist

After być:

• być fryzjerem
• być dentystą

This is a very common pattern:

• Jest nauczycielem. = He is a teacher.
• Chce być lekarzem. = He wants to be a doctor.

So the endings -em and -ą show the instrumental case.

Because dentysta is one of those masculine nouns that end in -a in the basic form.

Its instrumental singular changes like this:

• dentysta → dentystą
• artysta → artystą
• kolega → kolegą

So although dentysta refers to a man in grammatical usage here, it declines like a noun ending in -a, and that gives you -ą in the instrumental singular.

Ani is used in negative contexts and means something like nor.

So:

• nie chce być fryzjerem ani dentystą
  means
• he does not want to be a hairdresser or a dentist / neither a hairdresser nor a dentist

Because the clause is already negative with nie, ani is the natural connector.

A very common alternative is:

• nie chce być ani fryzjerem, ani dentystą

That version is a bit more symmetrical and emphatic, but the sentence you were given is also natural.

For the same reason as chce być: after woleć = to prefer, Polish also uses an infinitive.

So:

• woli pracować = prefers to work
• woli zostać w domu = prefers to stay at home

Also, pracować is imperfective, which fits here because the sentence is talking about work in a general or ongoing sense, not one finished action.

Because w meaning in/at a location usually requires the locative case, and the noun changes form.

So:

• basic form: metro
• locative after w: w metrze

Other examples of the same idea:

• w domu = in the house
• w sklepie = in the shop
• w tunelu = in the tunnel

So w metrze means in the subway / in the metro.

Przy usually means by, near, or at.

In a work context, it can also mean working by something or working on something connected with it.

So pracować przy nowym tunelu suggests work connected with the new tunnel project or work physically near it. It does not necessarily mean being inside the tunnel.

That makes przy slightly broader than a simple one-word English translation.

Because przy requires the locative case, and both the adjective and noun have to match that case.

So:

• nowy tunel = a new tunnel
• przy nowym tunelu = by/at the new tunnel

Changes:

• nowy → nowym
• tunel → tunelu

This is normal adjective-noun agreement in the locative case.

The semicolon links two closely related full clauses:

• Mój brat nie chce być fryzjerem ani dentystą
• woli pracować w metrze przy nowym tunelu

It is stronger than a comma but weaker than a full stop. In English and Polish, it can be used when two complete ideas are tightly connected.

You could also write this differently, for example with a full stop, or with a conjunction such as a in some contexts, but the semicolon is perfectly natural here.