When a priest celebrates Mass, he has at least two specific intentions.  The first intention, that he offers privately, is that he will celebrate the Mass well, according to the rubrics provided by the Church in the Roman Missal.  The second is a public intention where he applies the grace of that Mass toward a specific need or intention.

Often when priests are vesting, they offer a quiet prayer signalling their intent and desire, to pray the Mass well.  The second intention, known as the Mass intention, is announced publicly through various church communication means and usually at some point during the Mass on the designated day.

Often, people request a Mass for a specific intention.  They may be for deceased family members and friends, or for special intentions for the living, such as a wedding anniversary or birthday. These special Mass intentions are offered to God as prayers of intercession and thanksgiving.

While a priest may only accept one intention at a single Mass, he may have other intentions, not attached to an offering, that he holds in his heart. In addition, the faithful may bring their own intentions to the Mass, which they each carry privately and place upon the altar spiritually in prayer.