There is no visa to South Ossetia, only a permission from the MFA that you are allowed to visit. Permission is granted based on the itinerary you submit, so you will get the amount of days needed to complete your itinerary. Once permission is granted, you will get an e-mail back.
There is no clearance letter like in Abkhazia, but border authorities will (hopefully) know you’re coming. There is no visa system, just a migration card given at entry. You need a multiple entry or double entry Russian visa in order to re-enter Russia. Entry from Georgia is not possible, see South Ossetia border crossings.
Previously, to apply for permission, you had only to send an e-mail to the authorities. Nowadays, authorities ask for a letter of invitation from a local. Where you may find such a letter of invitation is still under discussion. You may weigh in at the South Ossetia visa forum topic.
Current options on the table are the Ministry of Culture of the Republic of North Ossetia-Alania (Vladikavkaz, ulitsa Revolutsyi, 61, works Tu-Su 10am-5.30pm) and the consular agency of SO in Vladikavkaz (it’s situated on ulitsa Krasodonskaia, 43, works Mo-Fri 9am – 6pm, lunch 1pm-2pm)
E-mail addresses to use to contact are [email protected] and [email protected] (and no longer [email protected]). More contacts can possibly be gleamed from the MFA website (hosted on the wonderful .su domain extension!).
For a feeling of what South Ossetia is like these days, read the South Ossetia trip report (2016).