Gemstones are a currency that can be used on your run to open chests that contain differents items, like gold or potions. You can also use them to buy Divine Graces from the Shopkeeper.
Gemstones may be gained from defeating enemies or founded on corpses, so be sure to check everywhere on each floor ! They drop randomly, but you can increase the rate at which gemstones are dropped by upgrading the "Gemstone Proficiency" stats of the Book of Rea.
Please note that unused gemstones disappear at the end of each dungeon, or when you die. So we advise you to spend them while you can, before fighting a boss for example!