'People's lives' is the plural form and is used when referring to the lives of multiple individuals. 'People's life' is the singular form and is used when referring to the collective life of a group of people or to people in general.

Because “Peoples” is usually singular, people’s’ is usually correct. We use peoples when we want to make the word “people” plural. This only happens in a situation where you want to show that there are different groups of people.

Remember, “people” as a word already refers to a group, so “people’s” looks at the collective possession within that single group, while “peoples'” expands this to include possessions or characteristics of multiple distinct groups.