Albert Barnes on Matthew 6:9

Our Father. God is called a Father,

(1.) as he is the Creator and the Great Parent of all.

(2.) The Preserver of the human family, and the Provider for their wants, Mt 5:45, 6:32.

(3.) In a peculiar sense the Father of those who are adopted into his family, who put confidence in him, who are true followers of Christ, and made heirs of life, Rom 8:14-17.

Hallowed be thy name. The word hallowed means, to render or pronounce holy. God's name is essentially holy; and the meaning of this petition is, "Let thy name be celebrated, and venerated, and esteemed as holy everywhere, and receive of all men proper honours." It is thus the expression of a wish or desire, on the part of the worshipper, that the name of God, or God himself, should be held everywhere in proper veneration.