What about our lives and words "hallows" the name of our Father?
In the book of Genesis, it states that we were created in The Father's image. As such, we are His representation on earth. Our beings, actions and words should glorify Him. As believers, we should be set apart from the world and be evidence of His glory. Every thought, movement or word "hallows" Him or at least should.
What desecrates and besmirches it?
Forgetting that we are His children and the evidence of His light upon the earth destroys not only His image, but also our faith. If we truly wish to "hallow" Him, then we need to be an example of Him with all we do. Our failure to live by His word doesn't just desecrate Him, but also destroys His image in the world. Being that the world is in a darkness and needs His light, we fall short of the image that He created us in when we fail to live by The Word and fail to let it be our guide.
How should we "hallow" the Father when we begin to pray?
It is said that when we study, God speaks to us and when we pray, we speak to Him. In our prayers, we are not just speaking to one of our friends in a casual manner, but we are speaking to Our Father, Our Creator and Our King. Our prayer should reflect this. We should come to Him with reverance and love that stands above all, for that is how He loves us.