Why is Satan called the prince of the power of the air?

The Navarre Bible New Testament¹ Compact Edition note on Eph 2:1-10 has:

[...]. The "prince of the power of the air" (v.2) refers to the devil as the leader of superhuman powers which, according to a widely held notion at the time, dwelt in the earth's atmosphere.

^{1. in the Revised Standard Version with a commentary by members of the Faculty of Theology of the University of Navarre.}

There are two words in Greek that could describe "air": οὐρανός (ouranos) and ἀήρ (aēr; obviously the origin of the English word). The latter word is what appears in Ephesians.

Whereas οὐρανός usually means "heaven" or "the heavens", ἀήρ refers to the space just above the surface of the earth ("sky"). Sometimes οὐρανός is used to mean "sky" (e.g. Matthew 13:32), but ἀήρ is never used to mean "heaven". It is a rare word and appears only 7 times in the New Testament, and 10 in the Septuagint.

In the Septuagint version of Genesis, οὐρανός is what appears in 1:1 (In the beginning God made the heaven and the earth). ἀήρ, however, is what is hovers over the earth. For example:

Psalm 17:11 LXX

And he made darkness his secret place: round about him was his tabernacle, even dark water in the clouds of the air [νεφέλαις ἀέρων].

The space between heaven and earth - the "sky" - is inhabited by principalities, powers, world rulers of darkness (Ephesians 6:12). John Chrysostom's explanation (4th c. Greek) for the wording here is:

Here again he means, that Satan occupies the space under Heaven, and that the incorporeal powers are spirits of the air, under his operation. For that his kingdom is of this age, i.e., will cease with the present age, hear what he says at the end of the Epistle; “Our wrestling is not against flesh and blood, but against the principalities, against powers, against the world rulers of this darkness;” (Eph. vi. 12.) where, lest when you hear of world-rulers you should therefore say that the Devil is uncreated, he elsewhere (Gal. i. 4.) calls a perverse time, “an evil world,” not of the creatures. For he seems to me, having had dominion beneath the sky, not to have fallen from his dominion, even after his transgression.¹

This "kingdom of the air" is a literal, not a figurative kingdom. When Christ is tempted by the devil in the wilderness, the devil offers him all the kingdoms of the world (Matthew 4:8; Luke 4:5). There is nothing in Scripture that insinuates that all this is not, in fact, Satan's to give. On the contrary: we have the testimony of John that all the world is under the power of the evil one (1 John 5:19)

^{1 Homily IV on Ephesians}

I would simply add to Mawia's anwer that the expression "the air" is a way of identifying the evil, sentient beings in the spiritual realm; namely, Satan and his minions, or demons.

The Hebrews believed in, and wrote about, three heavens, and in English we would call them "the firmament," "space," and "the home of God" (or "the highest heaven," as in "God resides in heaven," or "God is enthroned in heaven"; see Isaiah 6:1-7).

Interestingly, the New Testament phrase "in the heavenlies" (or "in the heavenly realms") describes the believers' position or standing IN Christ. We are seated in the heavenlies with Him by faith at the right hand of God (see Hebrews 8:1; 12:2). Positionally--that is, in a judicial sense--believers in the eyes of God are IN Christ (see Ephesians 1:3,20; 2:6; 3:10; 6:12; and 2 Corinthians 5:17).

By contrast, Satan and his demonic forces, referred to variously in the New Testament as principalities, powers, rulers, authorities, world forces of darkness, and spiritual forces of wickedness in heavenly places (see Ro 8:38; Eph 6:12; Col 1:16; 2:15; 1 Pet 3:22) occupy, in a sense, the first heaven, although we see the evidence of their work vey clearly on earth, in and through people and perhaps even through some acts of nature (which we sometimes, perhaps mistakenly, call "acts of God"!).

This evidence takes many forms, and not just in obvious ways such as violence, wars, genocide, widespread injustice, and sexual immorality, but also in not-so-obvious ways, as in godless ideologies and worldviews, corrupt and repressive governments, and much more.

Fallen humankind has its own role in the creation and maintenance of wickedness, too. We cannot claim as comedian Flip Wilson did years ago, "The devil made [us] do it"! There is an expression "to be in league with the devil," which encapsulates quite well how both Satan (or the devil) and humankind each assumes a degree of culpability, though Satan assumes more, since he instigated the rebellion against God and exerts more influence than we do within the spiritual realm (see Matthew 18:7). Although Satan is a powerful enemy, we are not to fear him (2 Timothy 1:7; 1 John 4:4). We are also to resist him and even on occasions rebuke him in the name of Jesus (see 1 Pe 5:8,9; Jas 4:7; Jude 9).

Moreover, Satan is well organized and systematic in his assault on anything and everything God calls good. Aided by a hierarchy of demons under his control, the hallmarks of his cosmos--the lust of the flesh, the lust of the eyes, and the pride of life (1 John 2:16)--are evident in every society and culture in the world: from the blatant consumerism and materialism of the minority world, to the oppressive political and religious regimes in the majority world, the latter of which are manifested in atheistic communism and radical Islam, for example.

Satan is alive and well on planet Earth. He is organized; his army of demons and unclean spirits is well staffed and well trained, with unwavering loyalty to their "prince; and his mission is clear: to kill, to steal, and to destroy anything and everything that God has made or is in the process of building, including the universal church of His Son and of our Lord and Savior, Jesus Christ (John 10:10).

Empowered both by Jesus' victory on the cross and by the Holy Spirit's empowering and channeling of the unwavering loyalty of God's ever-expanding family on earth, Satan is a defeated foe (see John 10:10; 1 John 3:8; and Revelation 12:10-12).

In conclusion, we thank God that Satan was dealt a death blow at the cross of Christ, which freed God to "wrap things up" by bringing history as we know it to a close in a way that is consistent with His eternal plan, the counsel of His will, and His holy, righteous, and just character. In so doing, God will eradicate completely Satan's strongholds in the world and then usher in a new heaven and a new earth where righteousness will reign supreme, where sin will never again intrude, and where Satan will never again spoil what God has called good (see Revelation 20:10 and 21:1).