N.B.: This information is specific to Our Lady of Lourdes in Raytown, Missouri, and comes from the Parish Bulletin of 4 January 2015. Specific practices may vary at other parishes.
“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.” — Pope Leo XIII, Mirae caritatis (1902)
An individual may ask a priest to offer a Mass for a specific intention for a number of reasons. This could be done, for example, in thanksgiving or in petition for a specific intention. It is also very common to ask for a Mass to be said for the repose of the soul of a deceased friend or family member.
When a priest reverently and validly offers Mass in accord with the norms of the Church, it is in union with and for the good of the entire Church. It can simultaneously be offered for specific intentions, such as the repose of a persons soul or for a specific intention, which need not be made public and could simply be “for the intentions of <Name>.”
A stipend is customarily paid to the priest for offering a Mass for a specific intention. This stipend is usually $10 and is used at the discretion of the priest for his own upkeep or for acts of charity in which he engages.
The process of having a Mass said for a specific intention is simple. Place a note that says something to the effect of “I would like to have a Mass said on or around <date> for <intention>. Enclosed, please find a stipend of <$##>.” in an envelope and put it in the collection basket, bring it to the office, or deliver it to Deacon Richard, who keeps the Mass schedule. If the date you have requested is open, your intention will be scheduled for that time. If not, it will be scheduled for the nearest open spot. (As you can imagine, Masses are schedule on a first-available basis, and once a Mass is scheduled it cannot be moved without a grave reason.)
If you have questions, please contact Deacon Richard either at firstname.lastname@example.org or at 816.679-8974. (And, please!, even though a stipend is given to the priest, please do not refer to this as “buying a Mass.” Masses cannot be bought or sold and Deacon Richard finds the phrase “buying a Mass” slightly more irritating than a jellyfish sting. Please don’t give Deacon Richard a rash.)