Person, Persons, People, Peoples

Can you please clarify the relation and differences between these nouns?

For example, is it proper to use "persons" instead of "people"? Are they the same? As I believe that "people" is plural, how come there exists the plural of the plural (=peoples)? Does it have any usage?

The dreaded Elements of Style have this to say on this matter:

The word people is not to be used with words of number, in place of persons. If of "six people" five went away, how many "people" would be left?

So at least in formal or technical discourse you might want to prefer "persons".

I have also observed that "persons" is generally and widely used in official documents, highway signs, etc. in Britain, Canada, and the U.S.

In informal contexts, however, "people" is probably preferable, especially as it will allow you to steer clear of the, er, people from all walks of life who will try to tell you that "persons" is not a valid word at all.

But as both your question and Strunk point out, you should avoid mixing them too much, because there is no exact correspondence between "person"/"persons" and "people"/"peoples".

The difference between "people" and "peoples" has been answered here: People or peoples when referring to an indigenous population?

Persons would be used as a the plural of person when you are not referring to a group collectively but to a collection of individuals. For example, "Many people like ice cream," but "Corporations are considered persons under the U.S. Consitution."

In British English at least, "persons" is generally considered a more formal word than "people", e.g. it is often used in formal announcements or notices.

"Persons wishing to carry on hand luggage should consult the steward beforehand"

See also Doug's answer for more considerations.