Mystery Discs (also called Disc Stones or Saucer Stones) are flat, circular tablets that contain a monster's genetic makeup inside. These were said to be created by Colona and entrusted to the Tochikans as a way to help them and give them companionship.
In the Anime
As you can imagine, there were originally not that many mystery discs. Each one contained the DNA of a basic breed of monster, but as monsters became more commonplace, their genetic information was combined to create many more types. Later, the Pendant scientists created their own man-made versions of the saucer stones, called tablets.
When the Phoenix and Moo battled during the Final War, the Final Gate turned all of the monsters originally generated from mystery discs back to their original forms.
In the Games
In most of the video games, these stones are represented by the CDs the player uses. Each CD contains information that helps you unlock a specific monster breed. Monster Rancher 1 and Monster Rancher 2 have a special bonus called Disc Chips - pieces of broken discs whose internal monsters have already been revealed. These can be used when combining for a specific monster to help you get a desired result.