Angel comes from the Greek word ἄγγελος (angelos). Greek doesn't get much easier than that.
ἄγγελος (angelos) = message or messenger. Or angel.
Angel or Messenger?
Of the 186 occurrences of ἄγγελος (angelos) in the King James Bible, 179 are translated "angel" and only 7 "messenger."
What about the Old Testament?
The LXX (Greek translation of the Hebrew) uses ἄγγελος (angelos) here, and the Hebrew uses an equivalent word מַלְאָךְ (malak), which is also translated either angel or messenger.
Reminds me of the angel of the LORD that went before the Israelites in the wilderness, which is also from מַלְאָךְ (malak) or ἄγγελος (angelos).
"Behold, I send an Angel before thee, to keep thee in the way, and to bring thee into the place which I have prepared." (Exodus 23:20, KJV)
How do they know whether to translate it "angel" or "messenger"?
It must be difficult to decide sometimes whether ἄγγελος (angelos) means "angel" or "messenger".
For example, I Corinthians 11 has a very difficult verse about head coverings:
"Therefore the woman ought to have power upon the head for the angels." (I Corinthians 11:11, Julia)
Most translations say "angel". A few put "messenger." I chose Julia Smith's literal Bible translation above.
The Literal Standard Version has:
"because of this the woman ought to have a token of authority on the head, because of the messengers;"
(I Corinthians 11:11, LSV)
See the post on Which Bible Translation is the closest to the original languages.
Wouldn't it be great if you could read the original language and know when the Greek is the same, even though the English is different?

