Here it all is in a nutshell…
There are three different types of intentions that a priest can offer at each Mass:
1) The Mass is offered for the universal Church (i.e., all the living and deceased members). Each and every Mass always includes this intention, and is reflected in the Eucharistic Prayer. Whereas this particular intention is mandatory, the following two are optional.
2) A Mass may also be offered for a “special intention”. Only one special intention may be offered per Mass, per priest (i.e., if additional priests are present concelebrating, each may offer a different special intention). The intention may be for a single individual (such as for the repose of someone’s soul or for the health of someone who is sick), or for two or more people who are grouped together by a common association (for example, for a married couple celebrating their anniversary, for a group of people on a pilgrimage together, for the victims of a natural catastrophe, etc.). A special intention can also be for an event (for example, a good harvest) or for a cause (for example, the Pro-Life Movement).
When someone arranges to “have a Mass said” or “have a Mass offered” for a particular intention, it is along the lines of what I have described here in #2. Many people give a “stipend” (i.e., a donation) for this, usually about five to ten dollars in the U.S. This is entirely optional. A priest cannot charge a fee for this, or for providing any of the sacraments. Once a priest tells someone that he will offer a Mass for a special intention (regardless of whether or not he received a stipend), he is morally obligated to either offer that Mass intention, or pass it along to another priest. If there was a stipend, and if he passes the Mass intention along to another priest, the stipend should likewise be passed along. A priest may agree to offer a special intention on a certain date, but he is not morally obligated to actually do it on that date (but he still has to do it at some point).
The special intention of a Mass may or may not be verbally announced during the Mass. Sometimes it is printed in the church bulletin. All this is up to the pastor. When a special intention of a Mass is announced, it is usually done so at the beginning of Mass and/or during the Prayers of the Faithful. None of this is actually covered by the rubrics of the Missal. Many priests will silently contemplate the special Mass intention at the Opening Prayer (during the pause that follows “Let us pray…”).
Along the lines of what DexUK pointed out, a pastor is obligated to offer (as the special intention) one Mass each Sunday and Holy Day of Obligation for his parishioners. The Latin term for this Mass is pro populo (“for the people”). If for some reason he is unable to (for example, he’s sick in bed) he can do this at a later time (even at a weekday Mass).
As a side note, donations to a priest associated with other events such as baptisms, weddings, funerals, etc. are sometimes called “stole fees” even though, technically speaking, it is not a “fee”. The Church today is trying to get away from terms such as “stipend” and “stole fee”, preferring the term “donation” (to emphasize that this is a voluntary offering).
3) By virtue of his office, a priest can include one or more other personal intentions at a Mass. They can be for himself, for others, for events, or for whatever. They do not have to be grouped together by a common association or such. When a priest tells you, “I’ll remember you (or your sick aunt, or your charity event, etc.) at Mass,” this is probably (but not necessarily) what he means. These sorts of Mass intentions are usually not announced. I know of a priest who keeps a list of personal intentions on a piece of paper in his shirt pocket and touches it while silently praying, “For these intentions” during the Mass. But there is no official way of doing this, and a priest may offer this third type of intention is whatever what he desires.
At this time allow me to offer an elaboration (and clarification) concerning one of my brother’s comments in Post #4. Is there some kind of difference or distinction in terms of the grace associated with each of these three types? To the best of my knowledge, the Church has never officially stated anything along these lines. There may or may not be some sort of distinction, but it has not been revealed to us. Therefore, the answer to this question would be, “God only knows.” Naturally, on our end of things all this should be contemplated recognizing that the spiritual benefit of a single Mass is infinite.
