The possessive case is a subset of the genitive case, which is one of the 4 main cases in modern English. It is the form used to indicate possession (i.e., ownership). It is usually created by adding 's to the word.
Carl's haircut (i.e., the haircut belonging to Carl - not a haircut by hairdresser Carl) / The man's coat (i.e., a coat belonging to a man - not a coat for a man) / the dog's dinner (i.e., the dinner of a specific dog - not a dinner fit only for a dog)
There are several other ways of forming the possessive case:
In the example above, Monet's is not in the possessive case but
the genitive case. This expression denotes that the
paintings were by Monet. It does not indicate that Monet owned them.