A library case is a box designed to hold a VHS tape. It is usually made of hard plastic but some (usually tapes containing feature films) are made of vinyl with padding inside. They may or may not have hubs inside to hold the tape spools in place. While this packaging was mainly used for VHS, smaller versions were sometimes used for audio cassettes.
Image without hubs: http://ep.yimg.com/ay/ldb-cdpackaging/clear-vhs-library-case-video-case-full-sleeve-no-hub-psv14-100-pcs-9.gif
Image with hubs: http://ep.yimg.com/ay/ldb-cdpackaging/100-pcs-vhs-library-case-with-hub-clear-full-sleeve-psv14hub-3.gif
Image of compact cassette-size case: http://auctions.c.yimg.jp/img339.auctions.yahoo.co.jp/users/0/4/8/5/teruakiteru_60-img553x600-1376211776swzxs741319.jpg