Learn Italian: Buongiorno signora, come sta? - Good morning, ma'am, how are you?

While essere is the normal verb for "to be", Italian uses the irregular verb stare (which literally means "to stay" or "to remain") specifically when asking or talking about how someone is feeling or doing. That is why come sta? is the standard way to ask "how are you?".

Why does the sentence use sta (the he/she form) if the speaker is talking directly to "you"?

The speaker is addressing the woman respectfully using the formal "you" (Lei). In Italian, the formal Lei always takes the exact same verb form as "he" or "she" (lui or lei). If you were talking to a close friend instead of a signora, you would use the informal "you" form and ask come stai?.

Why isn't the actual word for "you" anywhere in this sentence?

Italian usually drops subject pronouns like Lei (formal you) or tu (informal you) because the verb form and the context make it obvious who you are talking about. You could theoretically say come sta Lei?, but simply asking come sta? is much more natural.

Does signora just mean "Mrs."?

It can mean "Mrs." when used right before a last name, like signora Rossi. However, when used all by itself to talk directly to a woman, as it is in this sentence, it translates respectfully to "ma'am" or "madam".