Saucer Stones (also called Disc Stones) are flat spherical stones that contain a monster's genetic information inside. They are man-made versions of the Mystery Discs that God originally created and gave to the Tochikan people. The Pendant civilization wanted a way to re-seal and commercialize monsters, and since there was no way to know what type of monster could be generated from a mystery disc, they "perfected" the process by making the saucer stones different from each other, using inscriptions to catalogue which type was in which disc (which is how you use inscription regeneration in the Advance and DS games).
Very few mystery discs still exist, but saucer stones have become the commonplace regeneration method. Many shrines keep them on-hand for registered breeders to use.