For the Basic/NTLM/file - prompt via form Results Authorization method, this setting controls what cookie type to use for the Search Appliance's copy of the user's credentials.
With Basic/NTLM/file - prompt via form set, when a user conducts a search for the first time, a form is presented (from the Search Appliance) asking for a user and password. The user/pass is sent back to the user as a cookie from the Search Appliance for use in future searches without having to re-prompt. The user/pass is also simultaneously used to validate search results via HTTP Basic/NTLM or Windows/SMB file access.
The Basic/NTLM/file Cookie Type setting controls whether this cookie from the Search Appliance should be Persistent (retained long-term - a year - so the user does not have to log in at each new session) or Session (discarded after browser closure for security).
Note that the cookie for Basic/NTLM/file Cookie Type is distinct from the Login Cookies; they are used for different access methods. The former originates from the Search Appliance and is only ever sent to/from the user and the Search Appliance: non-cookie-based access methods are then used from the Search Appliance to the result URLs for actual authentication. Login Cookies, however, originate from a third-party form-based login system, and pass from the login server to the user to the Search Appliance to the result URLs.