A pronoun (a word that stands in for another noun known by context) which implies possession or association to something else.

Possessive pronouns:
his, her, their, my, your, our, its

Not possessive pronouns!
he's, they're, you're, it's

Not even words!
her's, their's, your's, our's

In English there are two sets of possessive words derived from pronouns. There are those that are used independently: mine, yours, hers, his, its, ours, theirs.

Mine is bigger than yours.

Can I have hers? I broke mine.

These are true pronouns. They function as a noun phrase (NP). The other set are those that are followed by the noun possessed: my, your, her, his, its, our, their.

My hovel is smaller than your mansion.

Though these are traditionally grouped with the first set as 'possessive pronouns', they are in fact more accurately determiners (like a, the, this, some, every, or in a looser terminology adjectives, and linguists often call them the possessive adjectives, not possessive pronouns. They are the possessive forms of the pronouns I, you, she etc. but do not themselves function as pronouns.

The same distinction exists in other languages: French adjectives/determiners mon, ma, mes vs pronouns mien, mienne. In some language they are definitely adjectives because they go with determiners: Italian la mia casa.