It is an ancient custom for a priest to dedicate a mass to a particular intention, usually for the welfare for a particular person such as someone who is ill or has died. It can of course be for other intentions, such as for victims of a natural disaster, that people respond to their vocations, etc. In my experience this intention is announced, or printed in a bulletin, both to encourage everyone participating in the mass to pray for that intention as well, and also to let anyone who may have requested that intention know that thier request is being fulfilled.
And if you request a priest to offer a mass for a particular intention, it is customary to make a donation at that time. I don’t know of any universal rule about this, but in most areas there are local customs, and sometimes rules determined by the bishop, conference, or a religious order regarding such donations and how they are utilized. Sometimes such offerings are used by the priest for his personal expenses – in some areas much of a priest’s income can come from this. In other areas such donations may be part of a fund that supports an organization. And in various areas there are customary amounts for such a donation.
This is my understanding of the usual meaning of offering a mass. Of course, you might offer to God your participation in a mass as a personal spiritual intention, as you would offer to God other prayers or good works. That would be an interior intention, known only to you and God unless you had some reason to discuss it with anyone. I assume that this is what you mean by offering your Communion.
