Listen up, people

 I tell you this with utter assurance: Every time you see peoples' it will be wrong.

Oh, all right, not every time. Just nine hundred ninety-nine times out of a thousand. 

People on nearly every occasion will be a noun singular in form with plural meaning: multiple human beings. 

One of Wilson Follett's crotchets in Modern American Usage may be recalled by older members of the assembly. He comes down hard on the distinction between people and persons, saying that "when we say persons we are thinking, or ought to be, of ones--individuals with identities; whereas when we say people we should mean a large group, an indefinite and anonymous mass." 

That distinction, what we ought to be thinking, was badly eroded sixty years ago and today is virtually gone. Bryan Garner, of course, recalls the distinction, but he is fully aware that it is pedantic and would produce sentences that sound strained to our ears. He says, "Twelve persons on the jury seems stuffy to many readers, and most native speakers of English (since about the mid-1970s) would say twelve people on the jury," which "has come to be viewed as the more natural phrasing." 

But there remains that thousandth instance in which peoples is a noun plural in form and plural in meaning: "a body of persons that are united by a common culture, tradition, or sense of kinship though not necessarily by consanguinity or by racial or political ties and that typically have common language, institutions, and beliefs" (thank you, Merriam-Webster). The United Nations is a place where the peoples of the planet are represented, and it is only in that context that you are ever likely to see the plural possessive peoples' used correctly. 

And the people said ...