An individual may ask a priest to offer a Mass for several reasons: for example, in thanksgiving, for the intentions of another person (such as on a birthday), or, as is most common, for the repose of the soul of someone who has died. One must never forget the infinite graces that flow from the Sacrifice of the Mass which benefit ones soul. Given this understanding, we can add some specifics. When a priest offers Holy Mass, he has three intentions: First, to offer the Mass reverently and validly in accord with the norms of the Church. Second, to offer the Mass in union with the whole Church and for the good of the whole Church. Third, to offer the Mass for a particular intention, such as the repose of the soul of someone who has died.
Therefore, the effects of the Mass bring certain benefits or fruits. The general fruits of the Mass are the effects upon the whole Church to the living faithful as well as the poor souls in purgatory. For this reason, in the Canon of the Mass (the Eucharistic Prayer), a special mention is made for both the living and the dead.
The special ministerial fruits of the Mass are applied to the particular intention of the Mass, i.e. "for whom the Mass is offered."