Special Breeds are rare monsters that are usually very difficult to obtain. Most are created when a specific CD (such as a particular PSX disc or music CD) is inserted at the shrine.
Note that the main breed must be unlocked in order to obtain the rare breed - for instance, in order to obtain a rare Dragon breed, you would first have to unlock the Dragon species. The game Dead or Alive is known to generate (/???) monsters, while Metal Gear Solid will (on monster rancher 2) Develop GabooSoldier, Nirvana's Nevermind generates Swimmer, and Beck's Mellow Gold generates Moo. These are confirmed only for the NTSC version; what CDs generate rare monsters in PAL versions will differ in most cases.