In Eastern-rite usage, the doxology is recited in the liturgy after the Lord's Prayer; this is probably how the phrase crept into Eastern Greek mediaeval manuscripts of the New Testament. These manuscripts are ultimately the ones on which the KJV was based. The KJV served as the Bible for English liturgy for a long time, thus becoming traditional for Protestants.
On the other hand, the doxology never made its way into the text of the Latin Bible, which was the basis of Western-rite (Roman Catholic) liturgy. So it is not understood as part of the Lord's Prayer proper in the Catholic Church.
Newer Protestant translations make use of a scholarly critical text of the New Testament. Biblical scholars stand in general consensus that the original Gospel of Matthew did not have the doxology. They believe this because the earlier Greek manuscripts don't have the phrase, and neither do early translations of the text, while there is a clear motivation for adding it in later Greek manuscripts. So the phrase is absent from the text of most modern translations (though it might be noted in a footnote).
Even though it wasn’t part of the original prayer, it’s such a favorite that I want to talk about it anyway. When we pray this phrase, we remind ourselves that when it's all said and done, it's all about God's kingdom, power, and glory anyway--and not about ours. We begin by praying for God's kingdom and God's will, and we end by reinforcing the idea. In Mark 8, Peter starts out by affirming Jesus' kingdom "You are the Messiah (v. 29 NLT)," but then turns around and reprimands Jesus for predicting the cross. In other words, Peter is all about the Messiah working for Peter's will, but not so keen on Jesus doing the will of God. In response, Jesus tells him that he was behaving as an agent of Satan, and that he is "seeing things merely from a human point of view, not from God’s (v. 33 NLT).”
When we pray the Lord's Prayer, from beginning to end, we remind ourselves that it starts with God's kingdom, power, and glory--and it ends the same way. When we come to church, too often we become like Peter, seeking after our own kingdom, power, glory, and will. When we do this, we become puppets of the devil, working from an earthly perspective and not from God's. Jesus gives a solution to this kind of selfishness in verses 34-36 (NLT:
“If any of you wants to be my follower, you must give up your own way, take up your cross, and follow me. If you try to hang on to your life, you will lose it. But if you give up your life for my sake and for the sake of the Good News, you will save it. And what do you benefit if you gain the whole world but lose your own soul?"
Praying the Lord's Prayer is a model for living the Christian life. It's letting go and letting God have control. We do this each day, no matter what challenges arise. We do it at church, in the office, at school, and in the community, wherever we live our life for God's kingdom, power, and glory--and not our own.