A rare card is a special item that can be found during a dungeon and allow you a random chance at bonus loot after the boss is defeated. Rare cards come in three colors and award a specific class of loot:
Completing a Very Hard Mode or Blood Mode dungeon with SSS Rank will award a bonus Gold Card, SS will award a Silver Card, and S will award a Bronze Card. Some ranks at other difficulties may also award a bonus Bronze Card.
Monsters and the Roulette can also drop extra cards. These are random, though gold cards and silver are much more rare than bronze.