A Mass Card is similar to any other greeting card, except that it indicates that the recipient will be remembered in prayer, normally as part of a church service. As the name suggests, it is primarily a Catholic custom.
Here's how it works: you buy the card from a church, and supply the church with the name of the recipient, and the reason for sending. The recipient will then be one of the people the priest asks his congregation to pray for in a future service. You get a nice (albeit religious) card to send to your friend, your friend gets a bunch of prayers said on his/her behalf, and the church gets your money. Everybody wins!
Of course, it helps if all three parties to this transaction are practising Catholics. Mass Cards are most commonly sent as sympathy cards in times of bereavement, and their design generally reflects this, but a very religious person might use Mass Cards to mark any occasion, such as birthdays, wedding, graduations etc.