In modern-day America, or pretty much anywhere in the Western Hemisphere, this phrase means that we wish God's Will to be done throughout the entire physical globe, or terre firma. However, when Jesus spoke these words to His disciples, who a long time ago, in the Eastern Hemisphere, they knew exactly what He was talking about!
Coupled together with the fact that Jesus was an Ancient Near Easterner, the History of Israel told the disciples that they should seek for His will to be done through His People, whether they be religious leaders ( heaven ) or just an average citizen of Israel. You may have heard that the phrase 'heaven & earth' is most often, in Scriptural use, indicative, not of the physical earth & sky ( heaven ), but of the People of God.
You can read for yourself Scriptures like Deuteronomy 4:26 and Psalm 76:8 for examples of this. 'The Song of Moses' in Deuteronomy 32 begins with 'Give ear, O heavens, and I will speak; and hear, O earth, the words of my mouth.'
In the previous phrase, 'Your will be Done' we saw how in the historical context of Jesus words, He reminded His disciples that Israel, under that first covenant, had failed to follow Yahweh's Dominion Mandate, Though His Will was done throughout their history, it was despite, not because of their obedience or disobedience to His commands.
The Law, as given on Sinai, and through Moses, was inscribed in the heavens but was not observed on earth. Jesus' model for prayer sought for the marriage of Heaven & Earth, bringing them together, inscribing His Law on the living hearts of His People ( Jeremiah 31:33 & Ezekiel 36:26 ), and not on dead tablets of stone.
'On Earth, as it is in Heaven', today communicates what should always be our prayer, that as we confess with our mouth, we would bring to action. As our cleansed conscience leads us in the way everlasting, our lives are transformed to His Will.
In Thankfulness, we praise the Maker of 'Heaven & Earth' and continually pray that we would, through the Strength He lends, always encourage our several members to fulfill His Will in our lives.
His Will is done, as we live our lives in Communion with Him & with the Body of His Son, 'On earth, as it is in heaven'!
Charles Haddon Shank