When a priest celebrates Mass, he offers each celebration of the Eucharist for a particular person, or intention. By doing so he applies special graces from God upon that person or intention.
Similar to how we are able to intercede for others by our personal prayers, the Church is able to intercede for us through the celebration of the Mass. However, since the Eucharist is the “source and summit of the Christian life” the Mass possess a power that our personal prayers do not.
The practice of offering Mass for particular intentions is an ancient one, dating back to the early Church.