A passcode is the eight-digit number that most cards have in the bottom-left corner on the card (certain cards, such as "Gate Guardian", do not have a passcode, and some cards instead have Limitation Text in this spot).
Passcodes, while serving no real purpose in the actual game, are used in some video games. By entering the passcode found on a given card (usually in a "Password" screen), the player may obtain that card for their virtual collection. This usually has various restrictions. In most games, this can only be done once per card. In some games entering the code will get the player the card for free, while in others the player must also pay in-game currency. In the recent games, players must also have the card they are trying to get in their card list, either by already owning at least one copy of the card, or owning most of the cards in the set the card is from.
Yu-Gi-Oh! The Duelists of the Roses uses its own passcodes for cards, separate from the passcodes printed on the physical cards; these often contain a mixture of numbers and uppercase letters instead of being just numbers.