Magic Stones and Magic Stone Switches are placed in various spots in cities and dungeons. You can collect Magic Stones by touching them, and by examining a Magic Stone Switch, you can start a challenge.

By examining a Magic Stone Switch, Magic Stones will appear, and a Challenge will begin.
By collecting all the Magic Stones within the time limit, you will clear the Challenge, and you will acquire those Magic Stones.

If time runs out, the Magic Stones will disappear, and you will fail the Challenge.
If you could not clear it, you can always retry the Challenge.

There is an NPC somewhere in a dungeon who will trade equipment for your Magic Stones, so be on the lookout.