Messiah comes from the Hebrew word mashiah, which means “the anointed one”.
This idea of anointing is seen all throughout both The Old Testament and The New Testament. There are many examples of God commanding different people to anoint kings, prophets, and priests in The Old Testament.
“Anointing” literally means “to smear with oil” (it comes from the Latin word unguere, meaning “to smear”). God ordered this ceremony of rubbing people with oil (apparently usually a special kind of oil for which he gives us the recipe in Exodus 30:22-33) many times, seemingly as a special ritual to show that someone or something is set apart for God.
This process was especially important for men of high standing, like men who were about to become king or about to become priests.
So, when we read in The Old Testament about how The Messiah is coming (think Daniel 9:25-26), we can know that “Messiah” is just being used as a catchall term to describe the one who is specially chosen and set apart for God. We know that this Messiah was Jesus, and even though He was anointed with oil at times (like Mary of Bethany anointing Jesus’ feet/smearing his feet with oil), He was called the Messiah because the term “anointed one” had just become a general term to describe someone set apart and chosen by God.
Now, that’s for the Hebrew.
When The New Testament was written in Greek, though, the authors properly translated the word mashiah into Greek as khristos, which also means “the anointed one”. This Greek term is where we get the English word “Christ”.
There are also two instances in the whole of The New Testament where the word “Messiah” is used, John 1:41 and John 4:25. In both of these verses, the author of John uses the word messiah because he’s recording what someone who spoke Hebrew or Aramaic was saying. He does also immediately follow it up both times by saying that “Messiah” is just the Hebrew word for “Christ” (in English “anointed one”).
Also, the word “Messiah” had a real interesting history. As it traveled through Greek and Latin, it became Messias. And this used to be the English word for “Messiah” too. But, when the authors of the Geneva Bible (one of the first English Bibles) were translating it into English, they decided to make the word look more Hebrew. So, they replaced the ‘s’ with an ‘h’, giving us our modern English word for Messiah.
I’m not entirely sure why most translators choose to not translate the words “Messiah” and “Christ”, but I hope that this is some useful information for you.