The object for which a priest offers the Eucharistic Sacrifice. This intention is distinct from the priest's decision to offer Mass, which is necessary for valid celebration. It is also distinct from the effects of the Mass which benefit those who take part in the sacrifice and attend the Mass. These are called the special fruits of the Mass and extensively are without limit, as are also the effects on the entire Church, called the general fruits of the Mass.
Mass intentions refer to the particular purpose for which a specific Mass is offered. This may be to honor God or thank him for blessings received. But technically a Mass intention means that the sacrifice is offered for some person(s) living or dead. Also called the application of a Mass, it pertains to the ministerial fruits of the Mass. These fruits are both extensively and intensively finite in virtue of the positive will of Christ. Other things being equal, the more often the sacrifice is offered the more benefit is conferred.
The intention for which a priest offers a Mass is determined either by the common law of the Church, or by specific precept, or, most often, by the intention of the donor of a Mass stipend, or by the priest's own devotion. Since it is not absolutely certain that the ministerial fruits of the Mass are limited, a priest may conditionally (if the one giving the stipend suffers no loss thereby) offer the Mass for several intentions. It is assumed that the priest does not intend by these second or third intentions to fulfill an obligation of justice by these conditional applications.