From my upcoming book, Frequently Misused and Misspelled Words and Phrases (and how to use them correctly).
Wrong: The many people of Africa offer diverse cultures.
Right: The many peoples of Africa offer diverse cultures.
Wrong: The incredibly huge number of peoples in India leads to economic hardship and health problems.
Right: The incredibly huge number of people in India leads to economic hardship and health problems.
Wrong: We’re hunting for three specific people who escaped from jail.
Right: We’re hunting for three specific persons who escaped from jail.
The word people refers to a group of human beings in a general sense. But a people is used for a body of people sharing a similar culture. When discussing multiple such bodies of people, use peoples. Persons is a plural of person, naturally, and refers to particular individuals, even if unidentified (as in “person or persons unknown”).