What does torri mean?

Torri is usually a nickname for Victoria*, which means "victory."

* The nickname Torri/Torey/Tory is also used for other names that contain the letters -tor-, such as Salvatore, Astor, Tordis, Torquil...

In Italian, torri means "towers" and torre/torri is the Italian name for the Chess peices known as the castles or rooks in English.