I will not provide a comprehensive explanation of the these three words in Eph 6:12 - such can be found in any good lexicon. Instead, I will provide a brief explanation of each to show their distinction from each other:
ἀρχή (arché)
[In English we get several words from this such as Archon, archaic, etc. The Latin equivalent word in principia from which English derives words like prince, principality, principal, etc.]
The essential meaning of this word is of one who is capable of initiating something as a ruler, hence the translations like, "Archons", "princes", "rulers", etc.
ἐξουσία (exousia)
This word simply means "authority"; when applied to a ruling person (as in Eph 6:12) it means one who has authority to say or do something as a ruler. Hence the obvious translation, "authorities".
κοσμοκράτωρ (kosmokratór)
This compound word consists of two parts:
- kosmos = world
- κράτωρ = strong/mighty ruler
Thus, the obvious meaning is "mighty world rulers".
Now, having said all this, I do not think Paul's use of these three specific words should be obsessed over - it is simply a list of the various names of various types of rulers that are found in any functioning government; in this case the spiritual government of demons "of this dark world and against the spiritual forces of evil in the heavenly realms".
That is, Paul is stating that there exists an unseen world of spiritual demons all of whose leaders/rulers are bent on opposing God and His people, Christians. What follows is far more important to the Christian:
Eph 6:13-17 - Therefore take up the full armor of God, so that when
the day of evil comes, you will be able to stand your ground, and
having done everything, to stand. Stand firm then, with the belt of
truth buckled around your waist, with the breastplate of righteousness
arrayed, and with your feet fitted with the readiness of the gospel of
peace. In addition to all this, take up the shield of faith, with
which you can extinguish all the flaming arrows of the evil one. And
take the helmet of salvation and the sword of the Spirit, which is the
word of God.
