Learn Hardcore Portuguese (Brazil): A senhora quer arroz com feijão ou peixe com batata? - Do you want rice and beans or fish with potato, ma’am?

Literally, senhora means lady or madam, but in sentences like this it functions as a formal pronoun. So the sentence is addressing a woman respectfully.

The a is the feminine singular definite article, so literally it is the lady, but in real usage it often just works as part of the polite form of address.

Why is the verb quer and not queres?

Because a senhora takes third-person singular verb forms, not second-person forms.

So:

A senhora quer... = polite/formal you want...
Você quer... = neutral/common you want...
Tu queres... = you want... with tu grammar

Even though a senhora refers to the person being spoken to, grammatically it behaves like she.

Is this sentence formal?

Yes, it is polite and somewhat formal.

Using a senhora is common:

when speaking to an older woman
in customer service
in restaurants, shops, clinics, etc.
whenever you want to sound respectful

A less formal version would be:

Você quer arroz com feijão ou peixe com batata?

And in many situations in Brazil, people would simply say:

Quer arroz com feijão ou peixe com batata?