What does a possessive apostrophe mean in English?

A possessive apostrophe means just that. It means that the apostrophe is indicating that that noun has ownership or possession, purpose or origin of the noun that comes after it.

Examples:

This is John's house.

John owns the house, therefore, it is John's house.

We went to the children's playground.

The playground intended for children.

The term 'possessive apostrophe' is used to distinguish the apostrophe from a contraction using an apostrophe.