There are two specific points in the Mass where offerings can be made. The first, as others have said, is at the Offertory prayers, where the priest holds up the bread and chalice. At this point, pray to put yourself and all of your fears, your worries, your attachments, your joys, and most importantly, your will, onto the paten and into the chalice and offer them to God in union with the sacrifice of His son during the Mass. Empty yourself to God and give Him everything, so that when you receive the Eucharist, you can be filled with His love. You then ask for an intention. For example, I say this prayer during the offertory as I use my imagination to place all of myself into the offering of the Mass:
The Sucipe (by St. Ignatius of Loyola)- Take Lord, and receive all my liberty, my memory, my understanding, and my entire will, all that I have and possess. Thou hast given all to me. To Thee, O lord, I return it. All is Thine, dispose of it wholly according to Thy will. Give me Thy love and thy grace, for this is enough for me.
The intention - Lord, I offer this Mass, as I offer all of the Masses in which I assist, for the sanctification of priests, especially my parish priests. [If I have an additional intention: I also offer this for the conversion of souls, etc.]
It was recommended to me by a priest to make a general intention at some point that all Masses attended would always have an intention, that way if you forget to make an intention at just the right moment, you still have an offering to give to God for every Mass. My general intention is for the sanctification of priests and strength for the holy mother Church.
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Parenthetical sidenote: A general intention can be made for many devotions. For example, I’ve made a general intention that I offer every rosary I say, in addition to any specific intentions I may have, for the conversion of the soul of one particular person. I’ve also made a general intention that all of the prayers in the Divine Office I pray are offered for an increase to vocations for priests and religious, especially to increase the vocations to the Carmelite order. In addition, when I say my morning offering, in addition to offering the day for the conversion of souls, in reparation for sin, and for the intentions of my friends and associates, I offer the day for the sanctification of priests. My priest says that this practice of general intentions is how he continues to pray for people that he hasn’t even seen or thought about in years - when people ask him to pray for them, he adds a general intention to one of his devotional prayers, so that their intentions will stay in his prayers.)
The second place that an intention can be made is when receiving communion. I always have the same intention prayer, which I discovered in a
devotional book about the Sacred Heart: My dearest Lord Jesus, I now receive this communion for love of Thee, in atonement for all of the coldness and all of the sin that thou has ever received in the sacrament of your love.
Here’s a couple of homilies that help describe the offertory devotion further:
