Our father who is in heaven, hallowed be your name. To “hallow” is to reverence or highly esteem, and God’s “name,” in the biblical understanding, is His essence or nature. This petition therefore asks that we properly appreciate God’s awesome nature, that we better understand the other party to our relationship.
Your kingdom come, your will be done. By asking that His “kingdom come,” we ask that we be as fully submitted to God as a loyal subject is to his monarch. By asking that His “will be done,” we ask for help organizing our affairs and actions according to the Father’s general precepts and in carrying out whatever specific tasks He gives us. Both requests ask God to help us align ourselves with Him.
Give us this day our daily bread. This asks for help in looking to God, not other sources, to fulfill our needs, so that we depend on God in the spirit of Matthew 6:33.
Forgive us our debts, as we also have forgiven our debtors. This asks for an incentive to, and help in, reflecting God’s extraordinarily gracious nature as we interact with others. We are asking to incorporate an aspect of God's nature in our own behavior, in the same way that children take on their parents’ mannerisms and long married spouses come to share common characteristics.
Do not lead us not into temptation, but deliver us from evil. By this petition we ask God to protect us from anything that interferes with our relationship with Him, so that nothing gets between us and Him.
In sum, each petition asks for something that increases or maintains our intimacy with God, something that brings or keeps us in close relationship with Him.