Jewel and Stone are required items for getting a card through Card Trader. In order to get those items, you need to exchange cards with the same rarity and same attribute as the ones that you want to acquire or meet "Stage" requirements.
Jewels and stones are one of the duel rewards. Basically, having a lots of duels and scoring high are ways to earn those items since you can get one reward by 1,000 score. For example four rewards appear when your score is 4,000 or more and less than 5,000.
You can convert your cards into jewels and stones at the Card Catalog. What trader item you get is based on attribute and rarity of a card you convert. For example, a normal rarity spell card can be exchanged with 100 gold and a Stone of Spell.
Jewels and stones are also reward for completing some missions, although you cannot obtain many of those items by doing that.