The possessive contraction 's indicates ownership. The preposition of the indicates a connection/relation but not necessarily ownership.

In the case of "Pirates of the Caribbean," "of" means "coming from" rather than "belonging to". For possession, 's is more commonly used than "of" -- as in "my son's homework" or "Fred's girlfriend." You can say "my uncle's friend" or "a friend of my uncle" and both would mean the same thing, but they answer slightly different questions. "Who is that person?" "A friend of my uncle." "Whose friend is that?" "My uncle's friend." Either answer is easily understood. Good luck with your studies!

Re: CHILDREN's The apostrophe --'s indicates Possession. The school is the possession of the Children. It is "their" school, the one they belong to. . Re: "The PirateS of the Carribean" indicates a plurality of Pirates. There is not one Pirate, but many Pirates. The pirates operate in the Carribean. Please note that something special happens with punctutation in regard to possession, when the word or name or pronoun ends in an S. For example, New Atheist author Richard Dawkins wrote a book, so we can refer to Dawkins' book. When the name ends in S, we do NOT write, Dawkins's book. (By the way, even the staff at Amazon.com gets this confused, and they DO employ the errant punctuation on their website, referring to "Dawkins's book". So, my name is Bruce Bain, and the way we would show possession of something in regard to my name would be like this: Bruce's book, or ...Bruce Bain's book. For an author like Richard Dawkins, we would write Dawkins' book.

You can say: Hats of Carol and Pat or Carol's and Pat's hats. Remarks of the judges or the judges' remarks. Note: Since the use of the apostrophe varies among writers, it is not possible to make a hard and fast rule about the apostrophe in singular words ending in s. Thus Hughes' poetry and Hughes's poetry are equally acceptable. Punctuate according to pronunciation. If you say "Hugheses"or "McCullerses, write "Hughes's"and "McCullers's." If you say "Hughes"poems or "McCullers"novels, write "Hughes'" and McCullers'." - Ref. Warriner's English Grammar and Composition 1986 Pg. 828