When discussing prayer with His disciples, our Lord said, “This is how you are to pray: ‘Our Father in heaven, hallowed be your name, your kingdom come, your will be done on earth as it is in heaven. Give us today our daily bread and forgive us the wrong we have done as we forgive those who wrong us. Subject us not to the trial but deliver us from the evil one’" (Matthew 6:9-13). A similar version is found in Luke 11:2-4. Both versions do not include the ending sentence found in the Protestant version, “For thine is the Kingdom, the power, and the glory now and forever."
The “For thine…" is technically termed a doxology. In the Bible, we find the practice of concluding prayers with a short, hymn-like verse which exalts the glory of God. An example similar to the doxology in question is found in David’s prayer located in I Chronicles 29:10-13 of the Old Testament. The Jews frequently used these doxologies to conclude prayers at the time of our Lord.
In the early Church, the Christians living in the eastern half of the Roman Empire added the doxology “for thine…" to the gospel text of the Our Father when reciting the prayer at Mass. Evidence of this practice is also found in the Didache (Teaching of the Twelve Apostles), a first century manual of morals, worship, and doctrine of the Church. Also when copying the scriptures, Greek scribes sometimes appended the doxology onto the original Gospel text of the Our Father; however, most texts today would omit this inclusion, relegate it to a footnote, or note that it was a later addition to the Gospel. Official “Catholic" Bibles including the Vulgate, the Douay-Rheims, the Confraternity Edition, and the New American have never included this doxology.
In the western half of the Roman Empire and in the Latin rite, we see the importance of the Our Father at Mass. St. Jerome (d. 420) attested to the usage of the Our Father in the Mass, and St. Gregory the Great (d. 604) placed the recitation of the Our Father after the Eucharistic Prayer and before the Fraction. In his Commentary on the Sacraments, St. Ambrose (d. 397) meditated on the meaning of “daily bread" in the context of the Holy Eucharist. In this same vein, St. Augustine (d. 430) saw the Our Father as a beautiful connection of the Holy Eucharist with the forgiveness of sins. In all instances, the Church saw this perfect prayer which our Lord gave to us as a proper means of preparing for Holy Communion. However, none of this evidence includes the use of the doxology.
Interestingly, the English wording of the Our Father that we use today reflects the version mandated for use by Henry VIII (while still in communion with the Catholic Church), which was based on the English version of the Bible produced by Tyndale (1525). Later in 1541 (after his official separation from the Holy Father), Henry VIII issued an edict saying, “His Grace perceiving now the great diversity of the translations (of the Pater noster etc.) hath willed them all to be taken up, and instead of them hath caused an uniform translation of the said Pater Noster, Ave, Creed, etc., to be set forth, willing all his loving subjects to learn and use the same and straitly [sic] commanding all parsons, vicars, and curates to read and teach the same to their parishioners." This English version without the doxology of the Our Father became accepted throughout the English speaking world, even though the later English translations of the Bible including the Catholic Douay-Rheims (1610) and Protestant King James versions (1611) had different renderings of prayers as found in the Gospel of St. Matthew. Later, the Catholic Church made slight modifications in the English: “who art" replaced “which art," and “on earth" replaced “in earth." During the reign of Edward VI, the Book of Common Prayer (1549 and 1552 editions) of the Church of England did not change the wording of the Our Father nor add the doxology. However, during the reign of Elizabeth I and a resurgence to rid the Church of England from any Catholic vestiges, the Lord’s Prayer was changed to include the doxology, and this version became the standard for English-speaking Protestants.
The irony of this answer is that some Protestants sometimes accuse Catholics of not being “literally" faithful to Sacred Scripture and depending too much on Tradition. In this case, we see that the Catholic Church has been faithful to the Gospel text of the Our Father, while Protestant Churches have added something of Tradition to the words of Jesus.