Learn Portuguese (Portugal): A senhora tem a sua chave aqui.

In Portugal, it is considered much more polite to address someone formally using O senhor (for a man) or A senhora (for a woman) instead of the direct word for you, você. It literally translates to "The lady", but it functions exactly like a formal "You".

Why is the verb tem used here? Doesn't that mean "he or she has"?

Yes, tem is the "he/she" form of the verb ter (to have). Because you are addressing the person respectfully as A senhora (The lady), the verb must match this third-person subject, even though you are speaking directly to them.

Why is there an a before sua chave?

In European Portuguese, you almost always put a definite article (like o or a, meaning "the") before a possessive word. Since chave (key) is a feminine word, we use a, making it literally "the your key".

Does sua mean "your" or "her"?

It can mean either! Sua is the possessive word for third-person subjects. Because formal "you" (A senhora) uses third-person grammar, sua can mean "your (formal)", "his", or "her". In this context, speaking directly to the lady, it means "your".