As I read the paper one morning recently, my eye fell on the word peoples’, and I let out one of those more-in-sorrow-than-in-anger sighs.
People is a plural word, and its plural possessive form is people’s. Other people’s money, other people’s houses, other people’s grammar.
It is a synonym of persons, pace Wilson Follett, who was insisting half a century ago that persons is the proper plural for a group of identifiable individuals, people for larger, general groups. That is, Bryan Garner says, badly dated pedantry, and it is completely acceptable for you to say, for example, that 130 people attended the hearing.
Now people is also a singular, in the sense of a nation or an ethnic group, “a people.” And its plural form is peoples, as when you would say that the United Nations attempts to represent all the peoples of the planet. Its plural possessive form is peoples’, a form I think you are rarely likely to be called upon to use.