Most of the time, people is the correct word to choose as a plural for person. Persons is archaic, and it is safe to avoid using it, except in legal writing, which has its own traditional language. Peoples is only necessary when you refer to distinct ethnic groups (for example, within the same region).
“People” vs. “persons” as plurals
Person and people both derive from Latin, but from different words. Person came from persona, which first meant “mask,” like that worn by an actor, but eventually came to mean “an individual human.” People, on the other hand, came from populus, which means “the people” in the sense of a group from the same nation, community, or ethnic group.
There was a time in history when it was put forward that grammatically, persons should be the preferred plural any time more than one person was referred to as a countable noun, and people should be preferred for uncountable nouns. That practice did not become standard, and nowadays, the plural persons is only considered correct in legal contexts and, occasionally, when deliberately referring to humans individually rather than collectively.
Six persons came dressed as either Princess Leia or Darth Vader, but the rest of the people were all wookies. (Acceptable because the persons’ individual choices are relevant to the context.)
Six people came dressed as either Princess Leia or Darth Vader, but the rest of the people were all wookies. (Equally acceptable.)
More people should recycle their paper to save trees.
Why don’t more people realize that capybaras are rodents?
Sixteen people protested on the White House lawn this Saturday.
“Persons” or “people” in formal legal writing
In the legal world, including law enforcement, persons is used regularly. It is helpful because nothing is collective where the law is concerned; individuals are prosecuted, not groups. Some legal expressions, such as persons of interest and missing persons, reflect this grammatical preference. Some people use persons in writing that is legal-sounding, but not strictly legal by definition, such as rules and public notices.
There are two persons of interest being questioned for the murder of John Doe.
The Douglas County Police Department has seventeen open missing persons cases.
No more than six persons can occupy the hot tub at any one time.
One political context in which persons is correct is in the expression displaced persons.
Using persons outside of legal contexts can sound unnecessarily affected. Avoid it completely in business contexts.
“People” vs. “peoples” for ethnic groups and nationalities
When you refer to the people of a single ethnic group or nationality, always use the word people.
Emmanuel Macron was elected by the people of France on May 7, 2017.
“We here highly resolve that government of the people, for the people, and by the people will not perish from the earth.” (Abraham Lincoln)
Peoples is only used in cases when it is necessary to distinguish between ethnic groups within the same geographical or cultural context.
The peoples of the world practice a wide variety of religions.
“Persons,” “people,” or “peoples”?
The plural of person should be people in the vast majority of contexts, although legalese uses the plural persons. Peoples should be reserved for instances where you are referring to more than one distinct ethnic group.