Question

Why do people use “peoples” and “persons” in some sentences? When is it right to use them in context?

Answer 1

No. Although both people and peoples are grammatically correct, they cannot be used interchangeably because they have different meanings.

Case 1: When you use it in the sense of one people, many peoples:
Here, people (singular) means a group of human beings belonging to a particular race/ethnicity. The plural of such groups is called peoples.
E.g. 
The different peoples of the world have very diverse traditions.The Ukrainians are a nice people.
Case 2: When you use it in the sense of one person, many people:
Here, people means a group of persons. This is the regular and most common usage of the word people.
E.g. 
There are more than a billion people in India