In its singular sense it refers to, e.g., a type / tribe / nation of people, so its plural form is peoples, e.g. the peoples of Africa. In this usage, the plural possessive is peoples'.
In its more common everyday sense it has no singular form; * it is often used as a plural for person, a synonym for persons; * its possessive form follows the "add apostrophe-s" convention, as though it were singular: people's.
"STYLE is knowing who you are, what you want to say, and not giving a damn." — Gore Vidal "STYLE is knowing what sort of play you're in." — Sir John Gielgud