What about our lives and words "hallows" the name of our Father?
From the beginning God has expressed His desire to have an intimate relationship with mankind. Not even original sin could keep God from loving us. Our actions, both visible and secret, are known to God. We must allow the Holy Spirit to teach us how to live our lives in a manner that will bring honor to our Father...our Lord.
What desecrates and besmirches it?
Any actions that deny His influence in the lives of His children.
How should we "hallow" the Father when we begin to pray?
When we pray we should acknowlege God's soverienty. We should praise Him for the mecry and grace He imparts to us. We should express our thankfulness for the sacrifice He made in order to redeem us from our sin.