Cards and figurines have a chance of being dropped by NPCs upon death. When they are displayed in a museum, the museum increases rank, attracting more visitors and thus more profit. Museum rank increases by a smaller amount from cards than it does from figures, however. They can't be bought from stores but can be sold in the player's shop.
Cards also have the additional property of being able to be put into decks when they are used. Decks, however, are very rare items, and as there is no way to remove cards from decks and no way of getting any use from cards in decks, it is a better idea to simply save cards and put them into a museum.