An exclusive card is a special card that is usually hard to find or has special artwork of an already existing card.
Such cards usually have a higher value on the marketplace and have special restrictions attached to them when used for print on demand.
It can only be printed once per year, and when your order is confirmed, its printing rights are used and it won't be available again until it resets every January 1st.
An exclusive card whose printing right is used cannot be sold over the marketplace or given to a friend until the next reset (either you sell, give, or reprint). So be extra careful if you give away your exclusive card to a friend so they can print it.
You can check the current printing rights of your card by looking at the card details or in your collection. The brown icon shows on the left how many printing rights you have compared to the number of cards in your possession. X / X means all printing rights are available, while 0 / X means you have used all the printing rights.
Note: An exclusive card on sale (but not yet sold) is also considered -1X. It's locked until it's sold or removed from sale.