When a priest offers the Holy Mass, he prays that the graces which come from the Mass be bestowed on someone in particular, such as a parishioner, family member, or departed soul.
The intention he prays for is usually one that has been requested by a parishioner.
The laity, can ask a priest to offer Mass for someone we are praying for, whether that person is someone who has passed away or who is still living. The intentions can vary: for conversion, for health, on birthdays or anniversaries.
But why ask a priest to offer Mass? Why not just pray for them ourselves?
It is good to do this because of the multitude of spiritual fruits gained from the Holy Sacrifice of the Mass. It is the greatest and most efficacious of all prayers, because in it, we are offering Christ Himself to His Father: the perfect Lamb of God, the supreme and most pleasing Sacrifice.