Messiah (Hebrew mashiach) means the anointed one. In Israel, kings, priests, and prophets were anointed with oil to set them apart for God's service — so the Messiah was the specially Anointed One, the promised King from David's line who would deliver God's people and reign for ever. Centuries of prophecy pointed forward to him.
The Greek translation of Messiah is Christos — Christ. So when Andrew tells his brother we have found the Messias, which is, being interpreted, the Christ, and when Peter confesses thou art the Christ, they are declaring Jesus to be the long-awaited Anointed One. Christ is not Jesus' surname but his title: Jesus the Messiah, the King the whole Old Testament was waiting for.
Messiah in Scripture
He first findeth his own brother Simon, and saith unto him, We have found the Messias, which is, being interpreted, the Christ.