Angels

Angel (αγγελος) means messenger. In some contexts these angelos are messengers of various entities, messengers of theoi, of daimones, or drakons, of some lord, or of some person... they're messengers. Not modern supernatural Angel-creatures.

archangel

ἀρχάγγελος (archángelos) - ruler or chief messenger

It is composed of:

• ἀρχ- (arch-) = chief, first, ruler, principal (from ἀρχή, “beginning, rule, authority”)
• ἄγγελος (angelos) = messenger, envoy

ἀρχάγγελος is simply an ἀρχ- compound: “chief messenger,” just as words like
ἀρχιερεύς mean “chief priest” and
ἀρχιτέκτων mean “master builder” or “chief craftsman.”

from archon

ἄρχων (archōn)

It comes from the verb:

• ἄρχω (archō) — to begin, to rule, to command, to govern.

In Classical Greek

An ἄρχων was an actual political office. In Athens, for example, the Nine Archons (ἐννέα ἄρχοντες) were the city’s chief magistrates.

New Testament occurrences

The noun ἀρχάγγελος is surprisingly rare. It appears only twice:

the Greek New Testament never says there are multiple ranks of archangels or gives a formal hierarchy. The later elaborate angelic hierarchies (such as those popularized by Pseudo-Dionysius) are centuries later than the New Testament.