Our Father in Heaven: address Him as Father but acknowledge His heavenly being so far above ours. Notice the Lord says "our" father, not "my" father. Our relationship with God is inseparable from our relationship with our brothers and sisters in Christ.
Holy be your name: we pray that God will maintain the sanctity of His name in the earth and especially in our lives.
Thy Kingdom come Thy will be done in earth as it is in heaven: we pray for the fulfillment of His avowed purpose to fill the earth with His Glory (see Numbers 14:21)See: : The Kingdom of God on Earth
Give us this day our daily bread: We acknowledge Him as the source of all goodness and sustenance in our lives. We include others in our prayer - note that again he says "us" and "our" rather than "me" and "my"
Forgive us our sins as we forgive those who have sinned against us: Notice that our relationship with God in the forgiveness of sins is directly dependent on the way we have treated others. We will only be forgiven if we ourselves forgive.
Lead us not into temptation but deliver us from evil: here he is talking about temptation and evil in the sense of trial and negative circumstances of life. God does not lead a person into sin (see James 1:13). Remember how Jesus prayed earnestly in the Garden of Gethsemane for the horror of the cross to be removed from him. The answer in this case, even when the son of God prayed for the burden to be removed, was "no." So with us, it is right to pray not to have to face trial and difficulty but sometimes, for our own good the answer to such prayer is "no." See Hebrews 12: 1-13.
For thine is the kingdom and the power and the glory for ever: He ends the prayer as he began it with an acknowledgement of God's will and glory to be fulfilled in God's Kingdom on earth.