Permesso di Soggiorno is a residency permit (aka permit of stay).
If you are staying in Italy longer than 3 months, you must apply for a residency permit. "Upon arrival in Italy, the American must obtain a permesso di soggiorno (permit of stay). As of December 11, 2006, an application 'kit' for the permesso di soggiorno may be requested from one of 14,000 national post offices (Poste Italiane). The kit must then be returned to one of 5,332 designated Post Office acceptance locations. It is important that applicants keep a copy of the receipt issued by the Post Office. Additional information may be obtained from an Italian immigration website via Internet at: http://www.portaleimmigrazione.it.
Within 20 days of receiving the permit to stay in Italy, Americans must go to the local Vital Statistics Bureau, Anagrafe of the Comune, to apply for residency. It generally takes one to two months to receive the certificate of residence (Certificato di Residenza)," explained the US Embassy.
Once you have applied, you may go to the Questura website (police department) with the file number (10 character code) or registered mail code (12 character code) to look up the status of your application.