Request Mass Intentions here 

Please keep in mind that the tradition of offering Masses for others, particularly the dead, originates in the very early Church. Given this understanding, we can add some specifics. When a priest offers Holy Mass, he has three intentions:

What does it mean to have a Mass "offered" for someone?

1. First, to offer the Mass reverently and validly in accord with the norms of the Church.
2. Second, to offer the Mass in union with the whole Church and for the good of the whole Church.
3. Third, to offer the Mass for a particular intention, such as the repose of the soul of someone who has died.

An individual may ask a priest to offer a Mass for several reasons:

In thanksgiving, for the intentions of another person (such as on a birthday on anniversary)
To have a Mass offered on the occasion of a birthday, anniversary or special need is appropriate, beneficial and appreciated.

To Intercede for another person in need
To offer a Mass for others who are are undergoing the challenges of life is a great gift.

For the repose of the soul of someone who has died.
When we face the death of someone, even a person who is not Catholic, to have a Mass offered for the repose of his soul and to offer our prayers are more beneficial and comforting than any other sympathy card or bouquet of flowers. 

Is there a fee for requesting a Mass intention? 
A person may ask a priest to offer a Mass for a particular intention; usually, a small stipend is given to the priest for offering the Mass, which thereby in justice creates an obligation which must be satisfied. 
The standard Mass Intention stipend is $10 but any offering is much appreciated.

Graces
One must never forget the infinite graces that flow from the Sacrifice of the Mass which benefit one’s soul. Pope Leo XIII in his encyclical "Mirae caritatis" (1902) beautifully elaborated this point and emphasized the connection between the communion of saints with the Mass:

"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." 

In his encyclical "Ecclesia de Eucharistia," our beloved late Holy Father, Pope John Paul II, taught, "

In the celebration of the Eucharistic Sacrifice, the Church prays that God, the Father of mercies, will grant His children the fullness of the Holy Spirit so that they may become one body and one spirit in Christ." (No. 43).

The special personal fruits of the Mass benefit the celebrating priest who acts in the person of Christ in offering the Mass and to the people who are in attendance and participate in the offering of the Mass. These fruits are both extensively and intensively finite, since each of us is finite. Therefore, the more a Mass is offered, the more benefit is conferred.