Jesus tells us that we can call the God of the universe, Father. He is our loving, gracious Father. Jesus then directs us to ask for God's name to be hallowed, or lifted up. I think this is an important part of the prayer that we sometimes forget in the course of our day-to-day. Our aim is to live our lives out in light of this prayer, rather than just repeating it from time to time.
If we pray for God's name to be hallowed, we are praying for that name to be made much of, to be lifted up and glorified. As the Westminster Catechism states, the chief end of man is to glorify God and enjoy him forever. The Psalmist tells us that creation itself declares the glory of the Creator. Everything and everyone is made to bring glory to God.
Sometimes we ask for God to be glorified, but we are not sure that we really want that. Sometimes God is glorified in things that we would consider tragedies. It is hard to look at the loss of a job, the death of a loved one, or anything else we would see as negative and see any way God can get glory through it. We can easily see glory for the Father when things go our way, but when it all hits the fan we act as if God's name is irrevocably damaged. I can think of many times when I questioned how in the world God could get any glory from things that happened.
It has been a hard lesson to learn, but it is true that the Father can be glorified through any and every thing. He is God, after all. Romans 8 tells us that God works in everything, even the messes in our lives, for the good of those who love him. This good could be directly for the person who is going through the stuff, or it could be for the body of Christ as a whole. Either way, God is glorified, whether it's through an individual becoming more like Christ or a group of Christians being strengthened by what the individual has gone through.
We are taught to pray that our Father be glorified. We need to understand that that glory may come through trouble in our lives. At the same time, we can be encouraged that the one who loves us with a perfect, inexhaustible love can and does redeem everything.