Like possessive nouns, a possessive pronoun owns or possesses. The manner in which possession is indicated varies with the type of pronoun: personal or indefinite.
Personal pronouns (e.g., I, he, and her) change case to show possession:

I: my or mine
us: our or ours
you: your or yours
he: his
she: hers
it: its
them: their

The possessive form of the pronoun "it" (its) is often mistakenly written "it's." It's" is a contraction for "it is."
Indefinite pronouns are made possessive by adding "'s":

To everyone's surprise, he arrived on time.
Please let me know if anyone's schedule changes.

When an indefinite pronoun followed by "else" is made possessive, apply the "'s" to the "else":

He hasn't sold anyone else's products in his store for ten years.
Margaret's clothes are unpacked, but everyone else's are still in the suitcases.