What does it mean to say that the Mass is offered for a person or persons?
The intention for the Mass refers to the priest's intention for the Mass. During the Liturgy of the Eucharist, the second half of the Mass that follows the universal prayer, the priest offers the Mass for the person or persons named in the last petition. While it is common to request Masses for the happy repose of the deceased, one may also have Masses said for those who are still living. Church law binds pastors to celebrate one Mass on Sunday for all of the people under his charge. This is not a Mass "for the parish" but rather "for the people"--as pastors sometimes have more than one parish entrusted to their care.
In order to request a priest to offer a Mass for a particular intention, one contacts the parish secretary. In the Diocese of Lafayette-in-Indiana, it is standard that one pay the priest $10 for the work of prayer that he performs. 1 Timothy 5:18b teaches us that "the laborer deserves his wages." The specific context here is the fact that priests deserve remuneration for the work that they do. Mass offerings make up just over 10% of a priest's annual compensation. I have decided to give Mass offerings to specific needs in the parish like rectory repairs or other items that could not otherwise fit in annual budgets. I find this to be an easy way to comply with the divine command to give 10% of one's income directly to God (see Malachi 3:10). Other priests depend on Mass stipends to pay bills or to help support needy family members.
While purchasing flowers is certainly a fitting way to honor deceased loved ones, offering Masses provides a more spiritually potent way to do so. For those of us preoccupied with the fate of living loved ones who are spiritually adrift, we can do no better than having Masses offered for their intention. I have witnessed more than once the positive effect that offering a Mass for a particular person has had for the person for whom it was celebrated.