Learn Russian: Каждый народ живёт по-своему.

It belongs to the group of adverbs you are learning right now, formed with the prefix по- and a special ending. Specifically, it comes from the reflexive possessive pronoun свой (one's own) in the dative case (своему). Together, по-своему translates to "in one's own way."

The translation says "Every people lives...", but народ is singular and takes the singular verb живёт. Why?

In Russian, народ means a "nation" or a distinct group of people sharing a culture or language, which is why it acts as a singular masculine noun here. When you want to say "people" as the plural of "person" (like "there are many people here"), you must use the plural word люди.

Why do we use по-своему instead of a form based on "its" (like его), since народ is the subject?

This is a great reminder of the reflexive possessive свой. Russian requires свой (and words built from it) whenever the "owner" is the subject of the sentence. Since народ is the subject doing the living, it lives in its own way (по-своему).

Why is there a hyphen in по-своему?