What is a Mass Card and How Are Mass Cards Used?
A Mass card, also known as a Mass offering card, memorial card or a remembrance card, is a greeting card given to someone to let them know that they, or a deceased loved-one, will be remembered and prayed for in the intentions at a Mass. The card-giver can obtain a Mass offering card from the parish office. If the card-giver goes directly to the parish office, they may obtain a card there, give his or her offering (a donation), and the Mass will be offered for his or her intention. He or she will then give the card to the recipient, which lets the recipient know that he or she or a deceased family member, or some other intention, will be prayed for at the Mass.

Why Offer a Mass for a Person or Intention?
"The grace of mutual love among the living, strengthened and increased by the sacrament of the Eucharist, flows, especially by virtue of the Sacrifice [of the Mass], to all who belong to the communion of saints. For the communion of saints is simply the mutual sharing of help, atonement, prayers and benefits among the faithful, those already in the heavenly fatherland, those consigned to the purifying fire, and those still making their pilgrim way here on earth. These all form one city, whose head is Christ, and whose vital principle is love. Faith teaches that although the august Sacrifice can be offered to God alone, it can nevertheless be celebrated in honor of the saints now reigning in Heaven with God, who has crowned them, to obtain their intercession for us, and also, according to apostolic tradition, to wash away the stains of those brethren who died in the Lord but without yet being wholly purified." - From the 1902 encyclical Mirae caritatis of Pope Leo XIII

Occasions for Offering Mass and Giving Mass Cards
The most common occasion for offering a Mass is for the repose of a departed soul. This tradition began quite early in the Church, as there are epitaphs on tombs as early as the 2nd century, which beg for prayers for the deceased person; and the Canons of Hippolytus, from the first half of the 3rd century, mention the offering of prayers for the dead at Mass. The saints and early church fathers spoke of remembering the dead at the Mass, and encouraged the faithful to pray for those who had died. Birthdays and anniversaries are also popular occasions for offering Mass, and any time you have a special intention to offer.  Healing Masses can also be offered for the intentions of those who are ill or recovering from illness or surgery.