In English, possessive words or phrases exist for nouns and most pronouns, as well as some noun phrases. These can play the roles of determiners (also called possessive adjectives when corresponding to a pronoun) or of nouns.
For historical reasons, this case is misleadingly called the possessive (case). It was called the genitive until the 18th century and in fact expresses much more than possession. Most disagreements about the use of possessive forms of nouns and of the apostrophe are due to the erroneous belief that a term should not use an apostrophe if it does not express possession.
In the words of the Merriam-Webster Dictionary of English Usage:
The argument is a case of fooling oneself with one's own terminology. After the 18th-century grammarians began to refer to the genitive case as the possessive case, grammarians and other commentators got it into their heads that the only use of the case was to show possession. ...
This dictionary also cites a study in whose samples only 40% of the possessive forms were used to indicate actual possession.
Nouns, noun phrases, and some pronouns generally form a possessive with the suffix -'s (apostrophe plus s, but in some cases just by adding an apostrophe to an existing s). Personal pronouns, however, have irregular possessives, and most of them have different forms for possessive determiners and possessive pronouns, such as my and mine or your and yours.
Possessives are one of the means by which genitive constructions are formed in modern English, the other principal one being the use of the preposition of.