This is answered by the Universal Catechism of the Catholic Church in its fourth major division on Christian Prayer under "The Lord's Prayer" (Part 3 of Article 4) in paragraphs 2786-2793.
http://198.62.75.1/www1/CDHN/pater1.html#FATHER
2786 "Our" Father refers to God. The adjective, as used by us, does not express possession, but an entirely new relationship with God.
2787 When we say "our" Father, we recognize first that all his promises of love announced by the prophets are fulfilled in the new and eternal covenant in his Christ: we have become "his" people and he is henceforth "our" God. This new relationship is the purely gratuitous gift of belonging to each other: we are to respond to "grace and truth" given us in Jesus Christ with love and faithfulness.[45]
2788 Since the Lord's Prayer is that of his people in the "endtime," this "our" also expresses the certitude of our hope in God's ultimate promise: in the new Jerusalem he will say to the victor, "I will be his God and he shall be my son."[46]
2789 When we pray to "our" Father, we personally address the Father of our Lord Jesus Christ. By doing so we do not divide the Godhead, since the Father is its "source and origin," but rather confess that the Son is eternally begotten by him and the Holy Spirit proceeds from him. We are not confusing the persons, for we confess that our communion is with the Father and his Son, Jesus Christ, in their one Holy Spirit. The Holy Trinity is consubstantial and indivisible. When we pray to the Father, we adore and glorify him together with the Son and the Holy Spirit.
2790 Grammatically, "our" qualifies a reality common to more than one person. There is only one God, and he is recognized as Father by those who, through faith in his only Son, are reborn of him by water and the Spirit.[47] The Church is this new communion of God and men. United with the only Son, who has become "the firstborn among many brethren," she is in communion with one and the same Father in one and the same Holy Spirit.[48] In praying "our" Father, each of the baptized is praying in this communion: "The company of those who believed were of one heart and soul."[49]
2791 For this reason, in spite of the divisions among Christians, this prayer to "our" Father remains our common patrimony and an urgent summons for all the baptized. In communion by faith in Christ and by Baptism, they ought to join in Jesus' prayer for the unity of his disciples.[50]
2792 Finally, if we pray the Our Father sincerely, we leave individualism behind, because the love that we receive frees us from it. The "our" at the beginning of the Lord's Prayer, like the "us" of the last four petitions, excludes no one. If we are to say it truthfully, our divisions and oppositions have to be overcome.[51]
2793 The baptized cannot pray to "our" Father without bringing before him all those for whom he gave his beloved Son. God's love has no bounds, neither should our prayer.[52] Praying "our" Father opens to us the dimensions of his love revealed in Christ: praying with and for all who do not yet know him, so that Christ may "gather into one the children of God."[53] God's care for all men and for the whole of creation has inspired all the great practitioners of prayer; it should extend our prayer to the full breadth of love whenever we dare to say "our" Father.
Paragraphs 2761 through 2865 (the very end of the Catechism) contain a full exposition of the Lord's Prayer, which is tremendously insightful, detailed, and powerful.
This is, by far, my favorite part of the Catechism.
My favorite paragraph of this section - because of its unique import is 2837, which is a mini-Bible study of the Greek word for "daily":
2837 "Daily" (epiousios) occurs nowhere else in the New Testament. Taken in a temporal sense, this word is a pedagogical repetition of "this day,"[128] to confirm us in trust "without reservation." Taken in the qualitative sense, it signifies what is necessary for life, and more broadly every good thing sufficient for subsistence.[129] Taken literally (epi-ousios: "super-essential"), it refers directly to the Bread of Life, the Body of Christ, the "medicine of immortality," without which we have no life within us.[130] Finally in this connection, its heavenly meaning is evident: "this day" is the Day of the Lord, the day of the feast of the kingdom, anticipated in the Eucharist that is already the foretaste of the kingdom to come. For this reason it is fitting for the Eucharistic liturgy to be celebrated each day.
The Eucharist is our daily bread. The power belonging to this divine food makes it a bond of union. Its effect is then understood as unity, so that, gathered into his Body and made members of him, we may become what we receive.... This also is our daily bread: the readings you hear each day in church and the hymns you hear and sing. All these are necessities for our pilgrimage.[131]
The Father in heaven urges us, as children of heaven, to ask for the bread of heaven. [Christ] himself is the bread who, sown in the Virgin, raised up in the flesh, kneaded in the Passion, baked in the oven of the tomb, reserved in churches, brought to altars, furnishes the faithful each day with food from heaven.[132]
