The English word Messiah comes from the Hebrew word Moshiach, which means Anointed One - in other words someone who has been especially chosen by God.|
In several religions anointing with oil is a way of making something holy and dedicating it to God. In Jewish history their kings (e.g. David, Solomon) were anointed with oil to bless their leadership - so Messiah also means King. Indeed the Queen was anointed at her coronation in 1953.
Greek was an international language used by many people in and around first century Palestine. The earliest Christians travelling and spreading their Good News about Jesus used Greek and the Christian New Testament was written in Greek. The Greek word for Anointed One is Christos, which is where we get the English word Christ.
So the words Messiah and Christ both mean the same thing: 'Anointed One' or 'King' - this is why Christians refer to Jesus as 'King of the Jews' and when they call him 'Jesus Christ' they mean Jesus the Messiah, Jesus the Anointed One or Jesus the King (so it's not a surname - as in son of Mr. & Mrs. Christ!).