Learn Italian: Come si chiama la tua insegnante?

In Italian, you don't usually ask what someone's name is using the verb "to be". Instead, you use the reflexive verb chiamarsi, which literally means "to call oneself". So Come si chiama translates word-for-word as "How does she call herself?".

Does Come mean "what" or "how" here?

Come literally means "how". Because Italians phrase this question as "How does she call herself?", we use Come instead of an Italian word for "what" (like Cosa). English translates the overall meaning as "What is she called?", but the Italian grammar strictly uses "how".

Why is there a la before tua? Doesn't tua already mean "your"?

Yes, tua means "your", but Italian almost always requires a definite article (the words for "the", like il or la) before a possessive word. You literally say "the your teacher". The main exception to this rule is with singular family members, where you drop the article (like tuo fratello for "your brother").

The word insegnante ends in -e. How do we know it is feminine here?