Catholics are used to "praying for the intentions of" somebody or something, as that is part of the Mass. But many non Catholics do not understand what "intentions" means, and Catholics themselves could use a refresher course, so here is brief explanation and a personal case study.
When one "prays for the intentions of" someone, or a group, or a cause, one is praying for the benefit of those people or causes. That is obvious because it is the same as praying "for" someone, which many people do frequently. But there is a second definition of intentions, which is more subtle, but also a firm statement of the power of the body of the Catholic communion's faith. The second definition is that when one prayers for the intentions of someone, you are not praying "for" them, but you are adding their prayers to whatever it is that the person is praying for. Thus, you are surrendering your "power of prayer" to add onto whatever the person is praying for, without necessarily knowing what that is. It is like the ultimate charity, such as in the traditional Jewish admonition that anonymous charity is even more blessed than charity for which one receives public or personal "credit." Here is an example of how it works.
Catholics regularly, in the Mass and privately, pray for "the intentions of the Pope." That is not praying for the Pope's personal benefit! So many non-Catholics misunderstand that. What we are doing is saying to God, "for whatever the Pope is praying for, I am adding my intentions and prayers to his." So when one prays "for the intentions of the Pope," at that moment one is adding one's prayers to whatever and whoever the Pope is praying for that day, at the Masses he celebrates and attends, or in the Pope's personal prayers. For example, if the Pope that day is praying for victims of a particular tragedy, and he might be doing so privately, in his own solo prayers, the faithful who pray for the intentions of the Pope on that day are "adding" their prayers onto the Pope's for the same "cause" and thus "intentions" of the Pope. This is why praying not for oneself but for the intentions of someone else is very powerful, because it is humble and therefore very pleasing to God. God is much more likely to answer prayers that are offered by his servants, such as the Pope, because the Pope constantly prayers on behalf of others, and not so much for himself. When one prays solely to "petition" (ask for) something specifically for one's self, that is the weakest form of prayer. But when one prays for others, especially in an "open ended" unconditional way in support of a servant of God, who is likewise, and in fact leading in the example of praying on behalf of others, great graces are bestowed.
By the way, I am not saying to not pray to God for a personal intention, far from it. But I am explaining that the less specific that you are with God and the more general is the goodness of your intention, the more grace and blessing you are likely to accrue, though perhaps not in the way you expected (and you may never know how effective it was, since the obvious thing was not granted by God, but something much better was, and often a disaster is avoided that one never knows about).
So here is an example of how I've prayed all my life with that spirit of intention. I tend to have a frequent "status of being in prayer," that is undetectable on the outside. What I mean is that like those who live in monasteries, I go about my work, but often with continual prayer on my mind. So the religious men or women of monasteries and nunneries are also "praying," even when they are tending the fields or doing other manual labor. I am like that. In fact, my list of intentions is often determined as things come to my attention during a normal day. Now I will give you a specific example.
When I lived in my house of twenty two years, I had a view from my hill of New York City, and the vast horizons of the sky to the north, east and south. Thus whenever I looked up in the sky I saw planes, since there were four airports within visual range of where I would stand. Commercial jets in particular flew low over my house when approaching Newark or New York City. I was often outside, either walking my dogs or working on the yard, and I always looked up whenever a plane flew overhead, and would try to see its colors and markings to determine what airline it was. And I always prayed for that plane's safety. Always.
In fact, the skies were so busy that I never saw less than four planes, which was especially easy to count at night. When I would be outside in the dark I always "inventoried" how many planes were in the sky that I could see by their moving lights. I always prayed for each one of them, and in fact, if someone could "read my mind" back then, they would have noticed that I fretted if I thought two planes were too close to each other (though it was usually due to optical illusion where they appeared near but were safely spaced in fact). So I also added what I could describe as "general aviation safety" prayers, hoping that everyone was doing their jobs and everyone would arrive safely.
So for decades I prayed for the safety of planes, their individual passengers, and the associated safety staff on a daily basis, often several times a day.
Now, I have to say, some odd things associated with flying occurred to me personally. For one, for years there was a helicopter that flew very low and aggressively over my house. There was no mistaking that this was, how to express it, some "optional" flying pattern, one that was deliberate and was not necessary. It would make the entire house rattle. Still, I never had a thought at that time that I was being deliberately harassed and disrespected.
Also I had some unpleasant personal flying experiences, one I won't go into details here, but they often had a "wrinkle" that was beyond the usual hardships of the traveling public (this was before the September 11 attacks). Again, I never associated any of them with my protective praying on behalf of air travelers in general. As mad as I may have gotten as a "consumer," I always maintained a gentle and prayerful heart toward those who travel by air, and continued to pray for those who I could see from my house.
However, shortly before the attacks of September 11, I was under siege myself. The cultists had made my life miserable in many ways, some of them I have documented here in this blog before.
Now, here is the problem. God does not expect people to pray when they are being tortured to do so. In other words, God does not expect people to pray for the goodness of the very people who are hurting you as they are doing it. And so I withdrew prayer intentions in general during that time, as I focused on trying to figure out what was going wrong with my world, and the world in general. I stopped having the daily reflexive and automatic praying on behalf of the fliers of the air, especially as I was being driven out of my home, and thus walking my dogs (all of whom had died but one) no longer was a pleasure where I counted and worried about the planes. I stopped praying not only for the daily intentions of air travelers, but also I, with God's fullness of knowledge, stopped my general intentions of praying for all other "groups" and intentions too. The only praying I have pretty much done since then is to add my intentions to that of the Pope's. However, once in a while I pray for those who are so blind and wicked that they cannot have the common sense to pray for themselves. That is an intention that I once in a while pursue. But again, God has let me know that I am not expected to "enable" people who are busy torturing and stalking me. So I've stopped praying for all groups, and this includes the innocent, such as infants and children, since they are part of the "to the third and fourth generation" of those who are at the very least gratifying their enabling cultist family members. Sadly, that is why my "intentions" oriented prayer life has disappeared, except when I attend Mass and join the intentions, and also I have an ongoing contribution to the intentions of the Pope (and I do pray for him, his health and his safety, personally, as I always have for all the Vicars of Jesus Christ).
You know what made me really sad (and disgusted?) Several years ago I read the "Left Behind" series, which if ever there should have been abortion on earth, it should have been abortion of those books. I've commented on them before. But I could not believe the hubris in the books related to the main characters being constantly in planes, being "pilot heroes" and so forth. How unconscious and ironic. A lot of pieces clicked into place for me then, but like I said, I had already, through my relationship with God, had reached the correct conclusions without knowing the specific "howes." But the arrogance of authors who condemn Catholics and Muslims as not being "saved" in the rapture, while drooling over the "fly boy" pilots who are the "heroes" during the "time of tribulation" was not lost on me, and I'm boggled that they have, apparently, no shame on the subject.
So it is a long and sad earthbound journey since I've gone from daily prayers for those who flew over my home to today, where the only people who merit prayers are those that I join with in the Holy Father's intentions in Rome.