Disk library is an optional field that records where, in your station's libraries, the disk is located. Since many stations separate their libraries by genre, library is often interchangeable with genre.
The Disk library input appears in the playlist entry form as a drop-down menu. You set up the menu items in Section 5 of the Station configuration page.
The information is useful to staff because it helps them locate music that another DJ has played.
When library corresponds to genre, as it often does, then it is useful for listeners too.
'Private' is a special value meaning that the disk belongs not to the station but to a DJ, and Spinitron keeps track of which DJ entered it so station staff can find out to whom the disk belongs.
Disabling this field prevents keeping track of library/genre or using it as a search criterion or in report generation.