Should people’s lives have an apostrophe?
“Improving people’s lives.” The word people is already plural so it only needs an apostrophe-s to become possessive. As Tina D mentions, if you are using the word peoples to refer to plural whole populations, then you can use peoples’ – but that is not the way the word would typically be used.
How do you use people’s correctly?
Both are correct: “People’s” is used when “People” is taken as a whole, as a singular noun showing a group of people. “Peoples’” is used when “Peoples” is taken as a plural noun showing various groups of people, not one group, but several groups of people, meaning plural of plural, Peoples.