Both Christ and Messiah mean anointed or anointed one.
The term Christ is an English derivative of the New Testament Greek word christos, which means "anointed." The equivalent Hebrew word in the Old Testament is mashiach. This term is transliterated in the King James New Testament as messias (John:1:41; 4:25), a word that has come down into modern English, including many Bible versions, as "messiah." Both Christ and Messiah mean "anointed" or "anointed one."
What was the significance of anointing? The Oxford Companion to the Bible states: "In the Hebrew Bible, the term is most often used of kings, whose investiture was marked especially by anointing with oil (Judg[es] 9.8-15; 2 Sam[uel] 5.3; 1 Kings 1.39; Ps[alm] 89.20 ...), and who were given the title 'the Lord's anointed' (e.g., 1 Sam[uel] 2.10; 12.3; 2 Sam[uel] 23.1; Ps[alm] 2.2; 20.6; 132.17; Lam[entations] 4:20)" (Bruce Metzger and Michael Coogan, editors, 1993, "Messiah," p. 513, emphasis added).
Anointing, this source tells us, "was widely practiced in the ancient Near East; the Amarna letters [on clay tablets found in central Egypt] suggest that anointing was a rite of kingship in Syria-Palestine in the fourteenth century BCE [B.C.], and ...[a story from the time of Judges] assumes its familiarity (Judg[es] 9:8, 15)" ("Anoint," p. 30).
Yet, as this and other sources point out, it was not only kings who were anointed in Scripture. Israel's high priests were anointed (Exodus:29:7; Leviticus:4:3, 5, 16), as were some prophets (1 Kings:19:16).
In biblical usage, anointing is an act of consecration —setting one apart for the holy work of God. It was symbolic of the pouring out of God's Spirit onto someone (compare Isaiah:61:1; Romans:5:5)—representing God's power and intervention to, in the cases cited, perform the duties of the office one was anointed to. Jesus Himself was "anointed...with the Holy Spirit and with power" (Acts:10:38).
The Jews of Jesus' day eagerly anticipated a specific prophesied figure referred to in several scriptures as the Messiah or Anointed One, a great King of the lineage of David who, by the power of God, would restore Israel and rule the world. Jesus of Nazareth was that Anointed One—and He will yet fulfill these prophecies.
What about the name "Jesus"? How did He receive this name, and what does it mean?
In Matthew 1 we find that Mary was discovered to be pregnant during her engagement to Joseph. Joseph was considering how to best handle the difficult situation.
"But while he thought about these things, behold, an angel of the Lord appeared to him in a dream, saying, 'Joseph, son of David, do not be afraid to take to you Mary your wife, for that which is conceived in her is of the Holy Spirit. And she will bring forth a Son, and you shall call His name Jesus, for He will save His people from their sins'" (Matthew:1:20-21).
The Greek name "Jesus" is a transliteration of the Hebrew name Yehoshua or Yeshua, the English form of which is "Joshua." This name literally means God is salvation. So the angel's message to Joseph was "You shall call His name ' God is salvation,' for He will save His people from their sins." That name tells us of Jesus' purpose in God's plan—that it is through Him that God carries out His plan to save humanity from death, giving us eternal life in His family.